Limits and an
Introduction to Calculus
12.1
Introduction to Limits
12.2
Techniques for Evaluating Limits
12.3
The Tangent Line Problem
12.4
Limits at Infinity and Limits of Sequences
12.5
The Area Problem
12
In Mathematics
If a function becomes arbitrarily close to a
unique number L as x approaches c from
either side, the limit of the function as x
approaches c is L.
In Real Life
David Frazier/PhotoEdit
The fundamental concept of integral calculus
is the calculation of the area of a plane
region bounded by the graph of a function.
For instance, in surveying, a civil engineer
uses integration to estimate the areas of
irregular plots of real estate. (See Exercises
49 and 50, page 897.)
IN CAREERS
There are many careers that use limit concepts. Several are listed below.
• Market Researcher
Exercise 74, page 880
• Business Economist
Exercises 55 and 56, page 888
• Aquatic Biologist
Exercise 53, page 888
• Data Analyst
Exercises 57 and 58, pages 888 and 889
849
850
Chapter 12
Limits and an Introduction to Calculus
12.1 INTRODUCTION TO LIMITS
What you should learn
• Use the definition of limit to
estimate limits.
• Determine whether limits of
functions exist.
• Use properties of limits and direct
substitution to evaluate limits.
Why you should learn it
The concept of a limit is useful in
applications involving maximization.
For instance, in Exercise 5 on page
858, the concept of a limit is used
to verify the maximum volume of
an open box.
The Limit Concept
The notion of a limit is a fundamental concept of calculus. In this chapter, you will learn
how to evaluate limits and how they are used in the two basic problems of calculus: the
tangent line problem and the area problem.
Example 1
Finding a Rectangle of Maximum Area
You are given 24 inches of wire and are asked to form a rectangle whose area is as large
as possible. Determine the dimensions of the rectangle that will produce a maximum
area.
Solution
Let w represent the width of the rectangle and let l represent the length of the rectangle.
Because
2w 2l 24
Perimeter is 24.
it follows that l 12 w, as shown in Figure 12.1. So, the area of the rectangle is
A lw
Formula for area
12 ww
Substitute 12 w for l.
12w w 2.
Simplify.
Dick Lurial/FPG/Getty Images
w
l = 12 − w
FIGURE
12.1
Using this model for area, you can experiment with different values of w to see how to
obtain the maximum area. After trying several values, it appears that the maximum area
occurs when w 6, as shown in the table.
Width, w
5.0
5.5
5.9
6.0
6.1
6.5
7.0
Area, A
35.00
35.75
35.99
36.00
35.99
35.75
35.00
In limit terminology, you can say that “the limit of A as w approaches 6 is 36.” This is
written as
lim A lim 12w w2 36.
w→6
w→6
Now try Exercise 5.
Section 12.1
851
Introduction to Limits
Definition of Limit
An alternative notation for
lim f x L is
Definition of Limit
If f x becomes arbitrarily close to a unique number L as x approaches c from
either side, the limit of f x as x approaches c is L. This is written as
x→c
f x → L as x → c
which is read as “f x approaches
L as x approaches c.”
lim f x L.
x→ c
Example 2
Estimating a Limit Numerically
Use a table to estimate numerically the limit: lim 3x 2.
x→2
Solution
Let f x 3x 2. Then construct a table that shows values of f x for two sets of
y
5
4
x-values—one set that approaches 2 from the left and one that approaches 2 from
the right.
(2, 4)
3
2
1
2
3
4
1.99
1.999
2.0
2.001
2.01
2.1
3.700
3.970
3.997
?
4.003
4.030
4.300
5
From the table, it appears that the closer x gets to 2, the closer f x gets to 4. So, you
can estimate the limit to be 4. Figure 12.2 adds further support for this conclusion.
−2
FIGURE
f x
x
−2 − 1
−1
1.9
x
f(x) = 3x − 2
1
12.2
Now try Exercise 7.
In Figure 12.2, note that the graph of f x 3x 2 is continuous. For graphs that
are not continuous, finding a limit can be more difficult.
Example 3
Estimating a Limit Numerically
Use a table to estimate numerically the limit: lim
x→0
lim f (x) = 2
y
5
f(x) =
.
Solution
Let f x xx 1 1. Then construct a table that shows values of f x for two
x→ 0
(0, 2)
x
x 1 1
x
x+1−1
sets of x-values—one set that approaches 0 from the left and one that approaches 0 from
the right.
4
3
x
1
−2
0.001
0.0001
0
0.0001
0.001
0.01
1.99499
1.99949
1.99995
?
2.00005
2.00050
2.00499
x
−1
−1
FIGURE
f x
f is undefined
at x = 0.
0.01
12.3
1
2
3
4
From the table, it appears that the limit is 2. The graph shown in Figure 12.3 verifies
that the limit is 2.
Now try Exercise 9.
852
Chapter 12
Limits and an Introduction to Calculus
In Example 3, note that f x has a limit when x → 0 even though the function is not
defined when x 0. This often happens, and it is important to realize that the existence
or nonexistence of f x at x c has no bearing on the existence of the limit of f x as
x approaches c.
Example 4
Estimating a Limit
Estimate the limit: lim
x→1
x3 x2 x 1
.
x1
Numerical Solution
Let f x x3 x2 x 1x 1. Then
construct a table that shows values of f x for two
Graphical Solution
Let f x x3 x 2 x 1x 1. Then sketch a graph of the
function, as shown in Figure 12.4. From the graph, it appears that as x
approaches 1 from either side, f x approaches 2. So, you can estimate
the limit to be 2.
sets of x-values—one set that approaches 1 from the
left and one that approaches 1 from the right.
0.9
0.99
0.999
1.0
f x
1.8100
1.9801
1.9980
?
x
1.0
1.001
1.01
1.1
?
2.0020
2.0201
2.2100
x
f(x) =
x3 − x2 + x − 1
x−1
y
lim f (x) = 2
x→ 1
5
f x
4
(1, 2)
3
2
From the tables, it appears that the limit is 2.
f is undefined
at x = 1.
−2
x
−1
1
2
3
4
−1
FIGURE
12.4
Now try Exercise 13.
Example 5
Using a Graph to Find a Limit
Find the limit of f x as x approaches 3, where f is defined as
f x 0,
x3
.
x3
2,
Solution
y
4
f (x ) =
Because f x 2 for all x other than x 3 and because the value of f 3 is immaterial,
it follows that the limit is 2 (see Figure 12.5). So, you can write
2, x ≠ 3
0, x = 3
lim f x 2.
3
x→3
The fact that f 3 0 has no bearing on the existence or value of the limit as x
approaches 3. For instance, if the function were defined as
1
x
−1
1
−1
FIGURE
12.5
2
3
4
f x 2,4,
x3
x3
the limit as x approaches 3 would be the same.
Now try Exercise 27.
Section 12.1
Introduction to Limits
853
Limits That Fail to Exist
Next, you will examine some functions for which limits do not exist.
Example 6
Comparing Left and Right Behavior
Show that the limit does not exist.
lim
x→0
x
x
Solution
f(x) =
2
x
x
x
1,
x
1
f(x) = 1
−2
x
−1
1
x
−2
x > 0
and for negative x-values
x
1,
2
f(x) = − 1
FIGURE
Consider the graph of the function given by f x x x. From Figure 12.6, you can
see that for positive x-values
y
x < 0.
This means that no matter how close x gets to 0, there will be both positive and
negative x-values that yield f x 1 and f x 1. This implies that the limit does
not exist.
12.6
Now try Exercise 31.
Example 7
Unbounded Behavior
Discuss the existence of the limit.
lim
x→0
Solution
Let f x 1x 2. In Figure 12.7, note that as x approaches 0 from either the right or the
left, f x increases without bound. This means that by choosing x close enough to 0, you
can force f x to be as large as you want. For instance, f x will be larger than 100 if
y
f(x) = 12
x
1
you choose x that is within 10 of 0. That is,
3
0 < x <
2
−2
x
−1
1
−1
FIGURE
12.7
1
10
f x 1
> 100.
x2
Similarly, you can force f x to be larger than 1,000,000, as follows.
1
−3
1
x2
2
3
0 < x <
1
1000
f x 1
> 1,000,000
x2
Because f x is not approaching a unique real number L as x approaches 0, you can
conclude that the limit does not exist.
Now try Exercise 33.
854
Chapter 12
Limits and an Introduction to Calculus
Example 8
Oscillating Behavior
Discuss the existence of the limit.
lim sin
x→0
y
Solution
Let f x sin1x. In Figure 12.8, you can see that as x approaches 0, f x oscillates
f(x) = sin 1
x
between 1 and 1. Therefore, the limit does not exist because no matter how close you
are to 0, it is possible to choose values of x1 and x 2 such that sin1x1 1 and
sin1x 2 1, as indicated in the table.
1
x
−1
1
1
x
x
sin
−1
FIGURE
1
x
2
1
2
3
1
2
5
1
0
2
5
2
3
2
?
1
1
1
Now try Exercise 35.
12.8
Examples 6, 7, and 8 show three of the most common types of behavior associated
with the nonexistence of a limit.
Conditions Under Which Limits Do Not Exist
The limit of f x as x → c does not exist if any of the following conditions are
true.
1.2
−0.25
0.25
−1.2
FIGURE
f(x) = sin 1
x
12.9
1. f x approaches a different number from the right side
of c than it approaches from the left side of c.
Example 6
2. f x increases or decreases without bound as x approaches c.
Example 7
3. f x oscillates between two fixed values as x approaches c.
Example 8
T E C H N O LO G Y
A graphing utility can help you discover the behavior of a function near the
x-value at which you are trying to evaluate a limit. When you do this, however, you
should realize that you can’t always trust the graphs that graphing utilities display.
For instance, if you use a graphing utility to graph the function in Example 8 over
an interval containing 0, you will most likely obtain an incorrect graph, as shown
in Figure 12.9. The reason that a graphing utility can’t show the correct graph is that
the graph has infinitely many oscillations over any interval that contains 0.
Section 12.1
Introduction to Limits
855
Properties of Limits and Direct Substitution
You have seen that sometimes the limit of f x as x → c is simply f c, as shown in
Example 2. In such cases, it is said that the limit can be evaluated by direct substitution.
That is,
lim f x f c).
Substitute c for x.
x→ c
There are many “well-behaved” functions, such as polynomial functions and rational
functions with nonzero denominators, that have this property. Some of the basic ones
are included in the following list.
Basic Limits
Let b and c be real numbers and let n be a positive integer.
1. lim b b
Limit of a constant function
x→ c
2. lim x c
Limit of the identity function
x→ c
n
3. lim x c
n
Limit of a power function
x→ c
n c,
n x 4. lim x→ c
for n even and c > 0
Limit of a radical function
For a proof of the limit of a power function, see Proofs in Mathematics on page 906.
Trigonometric functions can also be included in this list. For instance,
lim sin x sin 0
x→ and
lim cos x cos 0 1.
x→ 0
By combining the basic limits with the following operations, you can find limits for a
wide variety of functions.
Properties of Limits
Let b and c be real numbers, let n be a positive integer, and let f and g be functions
with the following limits.
lim f x L
x→ c
and
1. Scalar multiple:
2. Sum or difference:
3. Product:
4. Quotient:
5. Power:
lim g x K
x→ c
lim b f x bL
x→ c
lim f x ± gx L ± K
x→ c
lim f xgx LK
x→ c
lim
x→ c
f x
L
,
gx K
lim f x n Ln
x→ c
provided K 0
856
Chapter 12
Limits and an Introduction to Calculus
Example 9
Direct Substitution and Properties of Limits
Find each limit.
tan x
x
a. lim x2
b. lim 5x
c. lim
d. lim x
e. lim x cos x
f. lim x 42
x→ 4
x→ x→ 4
x→3
x→ x→9
Solution
You can use the properties of limits and direct substitution to evaluate each limit.
a. lim x2 42
x→ 4
16
b. lim 5x 5 lim x
x→4
Property 1
x→4
54
20
tan x
x→ x
c. lim
lim tan x
x→ Property 4
lim x
x→ 0
0
d. lim x 9 3
x→9
e. lim x cos x lim x lim cos x
x→ x→ x→ Property 3
cos f. lim x 42 lim x lim 4
x→3
x→3
2
x→3
Properties 2 and 5
3 42
72 49
Now try Exercise 47.
When evaluating limits, remember that there are several ways to solve most
problems. Often, a problem can be solved numerically, graphically, or algebraically.
The limits in Example 9 were found algebraically. You can verify the solutions
numerically and/or graphically. For instance, to verify the limit in Example 9(a)
numerically, create a table that shows values of x 2 for two sets of x-values—one set that
approaches 4 from the left and one that approaches 4 from the right, as shown below.
From the table, you can see that the limit as x approaches 4 is 16. Now, to verify the
limit graphically, sketch the graph of y x 2. From the graph shown in Figure 12.10,
you can determine that the limit as x approaches 4 is 16.
y
16
(4, 16)
12
y = x2
8
4
−8
x
−4
4
−4
FIGURE
12.10
8
12
x
3.9
3.99
3.999
4.0
4.001
4.01
4.1
x2
15.2100
15.9201
15.9920
?
16.0080
16.0801
16.8100
Section 12.1
Introduction to Limits
857
The results of using direct substitution to evaluate limits of polynomial and rational
functions are summarized as follows.
Limits of Polynomial and Rational Functions
1. If p is a polynomial function and c is a real number, then
lim px pc.
x→ c
2. If r is a rational function given by rx pxqx, and c is a real
number such that qc 0, then
lim r x r c x→ c
pc
.
qc
For a proof of the limit of a polynomial function, see Proofs in Mathematics on
page 906.
Example 10
Evaluating Limits by Direct Substitution
Find each limit.
a. lim x2 x 6
b. lim
x→1
x→1
x2 x 6
x3
Solution
The first function is a polynomial function and the second is a rational function
with a nonzero denominator at x 1. So, you can evaluate the limits by direct
substitution.
a. lim x2 x 6 12 1 6
x→1
6
b. lim
x→1
x2 x 6 12 1 6
x3
1 3
6
2
3
Now try Exercise 51.
CLASSROOM DISCUSSION
Graphs with Holes Sketch the graph of each function. Then find the limits of
each function as x approaches 1 and as x approaches 2. What conclusions can
you make?
a. f x! " x ! 1
b. g x! "
x2
x
1
1
c. h x! x3
2x2 x ! 2
x
3x ! 2
2
Use a graphing utility to graph each function above. Does the graphing utility
distinguish among the three graphs? Write a short explanation of your findings.
858
Chapter 12
Limits and an Introduction to Calculus
EXERCISES
12.1
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1. If f x becomes arbitrarily close to a unique number L as x approaches c from either side, the _______
of f x as x approaches c is L.
2. An alternative notation for lim f x L is f x → L as x → c, which is read as “f x _______ L as x _______ c.”
x→ c
3. The limit of f x as x → c does not exist if f x _______ between two fixed values.
4. To evaluate the limit of a polynomial function, use _______ _______.
SKILLS AND APPLICATIONS
5. GEOMETRY You create an open box from a square
piece of material 24 centimeters on a side. You cut equal
squares from the corners and turn up the sides.
(a) Draw and label a diagram that represents the box.
(b) Verify that the volume V of the box is given by
(c) The box has a maximum volume when x 4. Use a
graphing utility to complete the table and observe the
behavior of the function as x approaches 4. Use the
table to find lim V.
x→ 4
3
3.5
7. lim 5x 4
x→2
1.9
x
V 4x12 x2.
x
In Exercises 7–12, complete the table and use the result to
estimate the limit numerically. Determine whether or not the
limit can be reached.
4
4.1
4.5
5
1.999
f x
2
2.001
2.01
2.1
1.001
1.01
1.1
3.001
3.01
3.1
1
0.999
?
8. lim 2x2 x 4
x→1
0.9
x
3.9
1.99
0.99
0.999
f x
1
?
V
(d) Use a graphing utility to graph the volume function.
Verify that the volume is maximum when x 4.
6. GEOMETRY You are given wire and are asked to form
a right triangle with a hypotenuse of 18 inches whose
area is as large as possible.
(a) Draw and label a diagram that shows the base x and
height y of the triangle.
(b) Verify that the area A of the triangle is given by
(c) The triangle has a maximum area when x 3 inches.
Use a graphing utility to complete the table and
observe the behavior of the function as x approaches 3.
Use the table to find lim A.
x→3
2
2.5
x→3
x3
x2 9
2.9
x
2.9
3
3.1
3.5
4
A
(d) Use a graphing utility to graph the area function.
Verify that the area is maximum when x 3 inches.
2.99
2.999
f x
10. lim
x→1
3
?
x1
x2 x 2
1.1
x
A 12x18 x2.
x
9. lim
1.01
1.001
f x
x
?
0.99
0.9
f x
11. lim
x→ 0
sin 2x
x
x
0.1
0.01
f x
x
f x
0.001
0
?
0.01
0.1
0.001
Section 12.1
12. lim
x→ 0
tan x
2x
x
31. lim
x→2
0.1
0.01
0
0.001
f x
x 2
0.01
f x
x1
x 2 2x 3
x 5 5
15. lim
x→ 0
x
x
2 3 4
14. lim
x→2
16. lim
sin x
x
1 e4x
24. lim
x→ 0
x
ln2x 1
25. lim
x→1
x1
lnx2
26. lim
x→1 x 1
35. lim 2 cos
x→ 0
4
−4
x
36. lim sin
x→1
x
2
y
2
x
−2
−3 −2 −1
1 2 3
12
x2
In Exercises 37–44, use a graphing utility to graph the
function and use the graph to determine whether the limit
exists. If the limit does not exist, explain why.
40. f x sin x,
41. f x 42. f x 12
x
4
8
lim f x
x→1
x 3 1
x4
x 5 4
x2
,
,
lim f x
x→ 4
lim f x
x→2
43. f x x1
,
x 2 4x 3
44. f x 7
, lim f x
x 3 x→3
8
4
x
1
−2
−3
3x 2
−8 −4
x
2
−2
1
16
x
−2
y
y
3 6 9
x
1
5
, lim f x
2 e1x x→ 0
38. f x ln7 x, lim f x
x→1
1
39. f x cos , lim f x
x x→ 0
y
−6 −3
2
37. f x x→2
15
12
9
6
y
4
3
In Exercises 29–36, use the graph to find the limit (if it
exists). If the limit does not exist, explain why.
x→4
1
x1
−2
−3
2
2
x 2
x > 2
30. lim
x→1
−1
In Exercises 27 and 28, graph the function and find the limit
(if it exists) as x approaches 2.
29. lim x 2 3
34. lim
y
x3
x→3
e2x 1
23. lim
x→0
2x
x2
x2 4
1 x 2
2x
22. lim
x→ 0 tan 4x
2x 1, x <
x 3, x 2x,
28. f x 2
x 4x 1,
x→2
x2
x2 5x 6
sin2 x
21. lim
x→ 0
x
27. f x 33. lim
−2
−3
3
2
1
1
x2 4
18. lim
x→2
x2
cos x 1
20. lim
x→ 0
x
x
2
x2
17. lim
x→4
x4
x→ 0
x
1
−2
−3
In Exercises 13–26, create a table of values for the function
and use the result to estimate the limit numerically. Use a
graphing utility to graph the corresponding function to
confirm your result graphically.
19. lim
3
2
1
−1
x→1
x1
y
3
2
0.1
13. lim
x 1
x→1
y
?
x
32. lim
x2
0.001
859
Introduction to Limits
lim f x
x→1
860
Chapter 12
Limits and an Introduction to Calculus
In Exercises 45 and 46, use the given information to evaluate
each limit.
lim f x 3,
45. x→c
lim gx 6
x→c
lim 2gx
(a) x→c
lim f x gx
(b) x→c
lim f x
(c) x→c
g x
lim f x
(d) x→c
lim f x 5,
46. x→c
lim gx 2
x→c
5gx
4f x
1
lim
(d) x→c
f x
In Exercises 47 and 48, find (a) lim f x!, (b) lim g x!,
(c) lim [ f x!g x!], and (d) lim [ g x!
x→2
x→2
gx x→2
x→2
f x!].
47. f x 48. f x x
, gx sin x
3x
2x2
x→2
x→2
about f 2? Explain your reasoning.
49. lim 10 x 50.
51. lim 52. lim x→5
x→3
2x2
4x 1
9x 53. lim x→3
1
lim x3
x→2 2
x→2
54. lim
x→5
x3
5x
6x 5
6
x2
x1
2x 3
3x
1
56. lim
57. lim
5x 3
x→2 2x 9
58. lim
59. lim x 2
3 x2 1
60. lim 55. lim
x→3
x2
x→1
5x
61. lim
x→7 x 2
63. lim e x
72. THINK ABOUT IT Use the results of Exercise 71 to
draw a conclusion as to whether or not you can use the
graph generated by a graphing utility to determine
reliably if a limit can be reached.
(b) If lim f x 4, can you conclude anything
In Exercises 49–68, find the limit by direct substitution.
2
(a) Use a graphing utility to graph the corresponding
functions using a standard viewing window. Do the
graphs reveal whether or not the limit can be
reached? Explain.
73. THINK ABOUT IT
(a) If f 2 4, can you conclude anything about
lim f x? Explain your reasoning.
x2 5
x3,
71. THINK ABOUT IT From Exercises 7–12, select a limit
that can be reached and one that cannot be reached.
(b) Use a graphing utility to graph the corresponding
functions using a decimal setting. Do the graphs reveal
whether or not the limit can be reached? Explain.
(a) x→c
lim f x gx2 (b) lim
6 f x gx
x→c
lim
(c) x→c
70. The limit of the product of two functions is equal to the
product of the limits of the two functions.
x→ 4
x2
x2 1
x→3
x
x→3
62. lim
x 1
74. WRITING Write a brief description of the meaning of
the notation lim f x 12.
x→5
75. THINK ABOUT IT Use a graphing utility to graph the
tangent function. What are lim tan x and lim tan x?
x→ 2
76. CAPSTONE Use the graph of the function f to decide
whether the value of the given quantity exists. If it
does, find it. If not, explain why.
y
(a) f 0
(b) lim f x
5
lim ln x
64. x→e
(c) f 2
lim sin 2x
65. x→
66. lim tan x
(d) lim f x
3
2
1
67. lim arcsin x
x
68. lim arccos
x→1
2
x→3
x→12
x→8
x4
x→ EXPLORATION
TRUE OR FALSE? In Exercises 69 and 70, determine
whether the statement is true or false. Justify your answer.
69. The limit of a function as x approaches c does not exist
if the function approaches 3 from the left of c and 3
from the right of c.
x→ 4
x→0
What can you say about the existence of the limit
lim tan x?
x→0
x→2
−1
x
1 2 3 4
77. WRITING
Use a graphing utility to graph the function
x2 3x 10
. Use the trace feature
given by f x x5
to approximate lim f x. What do you think lim f x
x→4
x→5
equals? Is f defined at x 5? Does this affect the
existence of the limit as x approaches 5?
Section 12.2
861
Techniques for Evaluating Limits
12.2 TECHNIQUES FOR EVALUATING LIMITS
What you should learn
• Use the dividing out technique
to evaluate limits of functions.
• Use the rationalizing technique
to evaluate limits of functions.
• Approximate limits of functions
graphically and numerically.
• Evaluate one-sided limits of
functions.
• Evaluate limits of difference
quotients from calculus.
Dividing Out Technique
In Section 12.1, you studied several types of functions whose limits can be evaluated
by direct substitution. In this section, you will study several techniques for evaluating
limits of functions for which direct substitution fails.
Suppose you were asked to find the following limit.
x2 x 6
x→3
x3
lim
Direct substitution produces 0 in both the numerator and denominator.
32 3 6 0
Why you should learn it
Michael Krasowitz/TAXI/Getty Images
Limits can be applied in real-life
situations. For instance, in Exercise
84 on page 870, you will determine
limits involving the costs of making
photocopies.
Numerator is 0 when x 3.
3 3 0
Denominator is 0 when x 3.
0
0,
The resulting fraction,
has no meaning as a real number. It is called an
indeterminate form because you cannot, from the form alone, determine the limit. By
using a table, however, it appears that the limit of the function as x → 3 is 5.
x
3.01
3.001
3.0001
3
2.9999
2.999
2.99
x2 x 6
x3
5.01
5.001
5.0001
?
4.9999
4.999
4.99
When you try to evaluate a limit of a rational function by direct substitution
and encounter the indeterminate form 00, you can conclude that the numerator and
denominator must have a common factor. After factoring and dividing out, you should
try direct substitution again. Example 1 shows how you can use the dividing out
technique to evaluate limits of these types of functions.
Example 1
Dividing Out Technique
Find the limit: lim
x→3
x2 x 6
.
x3
Solution
From the discussion above, you know that direct substitution fails. So, begin by factoring
the numerator and dividing out any common factors.
lim
x→3
x2 x 6
x 2x 3
lim
x→3
x3
x3
lim
x→3
x 2x 3
x3
Factor numerator.
Divide out common factor.
lim x 2
Simplify.
3 2 5
Direct substitution and simplify.
x→3
Now try Exercise 11.
862
Chapter 12
Limits and an Introduction to Calculus
The validity of the dividing out technique stems from the fact that if two functions
agree at all but a single number c, they must have identical limit behavior at x c. In
Example 1, the functions given by
f x x2 x 6
x3
gx x 2
and
agree at all values of x other than x 3. So, you can use gx to find the limit of f x.
Example 2
Dividing Out Technique
Find the limit.
lim
x→1
x1
x3 x 2 x 1
Solution
Begin by substituting x 1 into the numerator and denominator.
110
3
Numerator is 0 when x 1.
2
1 1 110
Denominator is 0 when x 1.
Because both the numerator and denominator are zero when x 1, direct
substitution will not yield the limit. To find the limit, you should factor the numerator
and denominator, divide out any common factors, and then try direct substitution again.
lim
x→1
x1
x1
lim
x3 x 2 x 1 x→1 x 1x 2 1
y
lim
x1
x 1x 2 1
Divide out common factor.
lim
1
x2 1
Simplify.
x→1
2
x−1
f (x ) = 3 2
x −x +x−1
(1, 12)
x
1
FIGURE
12.11
x→1
f is undefined
when x = 1.
2
Factor denominator.
1
12 1
Direct substitution
1
2
Simplify.
This result is shown graphically in Figure 12.11.
Now try Exercise 15.
In Example 2, the factorization of the denominator can be obtained by dividing by
x 1 or by grouping as follows.
x3 x 2 x 1 x 2x 1 x 1
x 1x 2 1
Section 12.2
Techniques for Evaluating Limits
863
Rationalizing Technique
You can review the techniques
for rationalizing numerators and
denominators in Appendix A.2.
Another way to find the limits of some functions is first to rationalize the numerator of
the function. This is called the rationalizing technique. Recall that rationalizing the
numerator means multiplying the numerator and denominator by the conjugate of the
numerator. For instance, the conjugate of x 4 is x 4.
Example 3
Rationalizing Technique
Find the limit: lim
x 1 1
x
x→ 0
.
Solution
0
By direct substitution, you obtain the indeterminate form 0.
x 1 1
lim
0 1 1
0
x
x→ 0
0
0
Indeterminate form
In this case, you can rewrite the fraction by rationalizing the numerator.
x 1 1
x
x 1 1
x
x 1 1
x 1 1
x 1 1
xx 1 1
x
x x 1 1
Multiply.
Simplify.
x
Divide out common factor.
x x 1 1
1
x 1 1
, x0
Simplify.
y
Now you can evaluate the limit by direct substitution.
3
x 1 1
lim
x
x→ 0
2
f (x ) =
1
1
1
1
1
11 2
x 1 1
0 1 1
1
f is undefined
when x = 0.
1
x→ 0
You can reinforce your conclusion that the limit is 2 by constructing a table, as shown
below, or by sketching a graph, as shown in Figure 12.12.
x+1−1
x
x
−1
FIGURE
(0, 12 )
lim
x
0.1
f x
0.01
0.5132
0.5013
0.001
0
0.001
0.01
0.1
0.5001
?
0.4999
0.4988
0.4881
2
12.12
Now try Exercise 25.
The rationalizing technique for evaluating limits is based on multiplication by a
convenient form of 1. In Example 3, the convenient form is
1
x 1 1
x 1 1
.
864
Chapter 12
Limits and an Introduction to Calculus
Using Technology
The dividing out and rationalizing techniques may not work well for finding limits of
nonalgebraic functions. You often need to use more sophisticated analytic techniques to
find limits of these types of functions.
Example 4
Approximating a Limit
Approximate the limit: lim 1 x1x.
x→ 0
Numerical Solution
Let f x 1 x1x. Because you are finding the
limit when x 0, use the table feature of a graphing
utility to create a table that shows the values of f
for x starting at x 0.01 and has a step of
0.001, as shown in Figure 12.13. Because 0 is
halfway between 0.001 and 0.001, use the average
of the values of f at these two x-coordinates to
estimate the limit, as follows.
lim 1 x1x "
x→0
Graphical Solution
To approximate the limit graphically, graph the function
f x 1 x1x, as shown in Figure 12.14. Using the zoom and trace
features of the graphing utility, choose two points on the graph of f, such as
0.00017, 2.7185
0.00017, 2.7181
as shown in Figure 12.15. Because the x-coordinates of these two points
are equidistant from 0, you can approximate the limit to be the average of
the y-coordinates. That is,
2.7196 2.7169
2.71825
2
The actual limit can be found algebraically to be
e " 2.71828.
and
lim 1 x1x "
x→ 0
2.7185 2.7181
2.7183.
2
The actual limit can be found algebraically to be e " 2.71828.
5
f(x) = (1 + x)1/x
−2
2
2.7225
−0.00025
12.13
FIGURE
12.14
0.00025
2.7150
0
FIGURE
FIGURE
12.15
Now try Exercise 37.
Example 5
Approximating a Limit Graphically
Approximate the limit: lim sin x.
x→ 0
x
f(x) =
2
−4
4
−2
FIGURE
sin x
x
12.16
Solution
0
Direct substitution produces the indeterminate form 0. To approximate the limit, begin
by using a graphing utility to graph f x sin xx, as shown in Figure 12.16. Then
use the zoom and trace features of the graphing utility to choose a point on each
side of 0, such as 0.0012467, 0.9999997 and 0.0012467, 0.9999997. Finally,
approximate the limit as the average of the y-coordinates of these two points,
lim sin xx " 0.9999997. It can be shown algebraically that this limit is exactly 1.
x→0
Now try Exercise 41.
Section 12.2
Techniques for Evaluating Limits
865
T E C H N O LO G Y
The graphs shown in Figures 12.14 and 12.16 appear to be continuous at x " 0.
However, when you try to use the trace or the value feature of a graphing utility
to determine the value of y when x " 0, no value is given. Some graphing utilities
can show breaks or holes in a graph when an appropriate viewing window is used.
Because the holes in the graphs in Figures 12.14 and 12.16 occur on the y-axis, the
holes are not visible.
One-Sided Limits
In Section 12.1, you saw that one way in which a limit can fail to exist is when a
function approaches a different value from the left side of c than it approaches from the
right side of c. This type of behavior can be described more concisely with the concept
of a one-sided limit.
lim f x L1 or f x → L1 as x → c
Limit from the left
lim f x L2 or f x → L2 as x → c
Limit from the right
x→c x→c Example 6
Evaluating One-Sided Limits
Find the limit as x → 0 from the left and the limit as x → 0 from the right for
2x
.
f x x
y
f(x) = 2
Solution
From the graph of f, shown in Figure 12.17, you can see that f x 2 for all x < 0.
Therefore, the limit from the left is
2
1
−2
2x
x
x
−1
1
−1
f(x) = − 2
f (x ) =
2
lim
x
Limit from the left: f x → 2 as x → 0
Because f x 2 for all x > 0, the limit from the right is
lim
x→0
FIGURE
2x
2.
x→0
2x
2.
x
Limit from the right: f x → 2 as x → 0
Now try Exercise 55.
12.17
In Example 6, note that the function approaches different limits from the left and
from the right. In such cases, the limit of f x as x → c does not exist. For the limit
of a function to exist as x → c, it must be true that both one-sided limits exist and
are equal.
Existence of a Limit
If f is a function and c and L are real numbers, then
lim f x L
x→c
if and only if both the left and right limits exist and are equal to L.
866
Chapter 12
Limits and an Introduction to Calculus
Example 7
Finding One-Sided Limits
Find the limit of f x as x approaches 1.
f x 44xx,x ,
2
x < 1
x > 1
Solution
Remember that you are concerned about the value of f near x 1 rather than at x 1.
So, for x < 1, f x is given by 4 x, and you can use direct substitution to obtain
lim f x lim 4 x
x→1
x→1
41
3.
y
7
For x > 1, f x is given by 4x x 2, and you can use direct substitution to obtain
f(x) = 4 − x, x < 1
6
f(x) = 4x −
5
lim f x lim 4x x2
x 2,
x>1
x→1
x→1
41 12
4
3.
3
2
Because the one-sided limits both exist and are equal to 3, it follows that
1
x
−2 −1
−1
FIGURE
1
2
3
5
6
lim f x 3.
x→1
The graph in Figure 12.18 confirms this conclusion.
Now try Exercise 59.
12.18
Example 8
Comparing Limits from the Left and Right
To ship a package overnight, a delivery service charges $18 for the first pound and $2
for each additional pound or portion of a pound. Let x represent the weight of a package
and let f x represent the shipping cost. Show that the limit of f x as x → 2 does
not exist.
$18, 0 < x
f x $20, 1 < x
$22, 2 < x
Overnight Delivery
Shipping cost (in dollars)
y
1
2
3
23
22
21
20
19
18
17
Solution
For 2 < x ≤ 3, f (x) = 22
The graph of f is shown in Figure 12.19. The limit of f x as x approaches 2 from the
left is
For 1 < x ≤ 2, f(x) = 20
lim f x 20
x→2
For 0 < x ≤ 1, f (x) = 18
whereas the limit of f x as x approaches 2 from the right is
x
1
2
3
Weight (in pounds)
FIGURE
12.19
lim f x 22.
x→2
Because these one-sided limits are not equal, the limit of f x as x → 2 does not exist.
Now try Exercise 81.
Section 12.2
Techniques for Evaluating Limits
867
A Limit from Calculus
In the next section, you will study an important type of limit from calculus—the limit
of a difference quotient.
Example 9
Evaluating a Limit from Calculus
For the function given by f x x 2 1, find
lim
h→ 0
f 3 h f 3
.
h
Solution
Direct substitution produces an indeterminate form.
lim
h→ 0
f 3 h f 3
3 h2 1 32 1
lim
h→ 0
h
h
9 6h h2 1 9 1
h→0
h
lim
6h h2
h→0
h
lim
0
0
By factoring and dividing out, you obtain the following.
lim
h→ 0
f 3 h f 3
6h h2
h6 h
lim
lim
h→ 0
h→0
h
h
h
lim 6 h
h→0
60
6
So, the limit is 6.
Now try Exercise 75.
Note that for any x-value, the limit of a difference quotient is an expression of the
form
lim
h→ 0
f x h f x
.
h
Direct substitution into the difference quotient always produces the indeterminate form 00.
For instance,
lim
h→0
For a review of evaluating
difference quotients, refer to
Section 1.4.
f x h f x f x 0 f x
h
0
f x f x
0
0
.
0
868
Chapter 12
Limits and an Introduction to Calculus
EXERCISES
12.2
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1. To evaluate the limit of a rational function that has common factors in its numerator and denominator,
use the _______ _______ _______ .
0
2. The fraction 0 has no meaning as a real number and therefore is called an _______ _______ .
3. The limit lim f x L1 is an example of a _______ _______ .
x→c
4. The limit of a _______ _______ is an expression of the form lim
h→0
f x h f x
.
h
SKILLS AND APPLICATIONS
In Exercises 5–8, use the graph to determine each limit
visually (if it exists). Then identify another function that
agrees with the given function at all but one point.
2x 2 x
5. gx x
−2
−2
x
4
−2
−2
4
−6
(a) lim gx
1 2x 3x 2
x→1
1x
14. lim
13. lim
x→2
(b) lim gx
(b) lim hx
(c) lim gx
(c) lim hx
x→2
x3 x
7. gx x1
6
18. lim
x4 1
x1
19. lim
x2 x 2
x2 3x 2
20. lim
x2 2x 8
x2 3x 4
23. lim
y
x 3 3
x
x→ 0
25. lim
2x 1 1
x→0
2
4
2
−2
x
−2
2
x
−2
2
4
−4
(a) lim gx
(a) lim f x
(b) lim gx
(b) lim f x
(c) lim gx
(c) lim f x
x→1
x→1
x→ 0
x→1
x→2
x→1
4
a3 64
a→4 a 4
5 y 5
y→ 0
4
2x2 5x 3
x3
x5 32
x2
y
y
x→3
17. lim
21. lim
x2 1
8. f x x1
x2 6x 8
x→2
x2
16. lim
x→1
x→3
7x
x2 49
t3 8
t2
x→ 0
x→1
x→7
15. lim
x→2
(a) lim hx
x→ 0
12. lim
t→2
x
2
x2 2x 3
x→1
x1
11. lim
2
4
10. lim
x→6
y
6
x6
x2 36
9. lim
x 2 3x
6. hx x
y
In Exercises 9–36, find the limit (if it exists). Use a graphing
utility to verify your result graphically.
27. lim
x
x 7 2
x3
1
1
x1
29. lim
x→0
x
x→3
1
1
x4 4
31. lim
x→0
x
sec x
33. lim
x→0 tan x
35. lim
x→ 2
1 sin x
cos x
x→1
x→4
7 z 7
22. lim
z
z→0
24. lim
x 4 2
x
x→0
26. lim
x→9
3 x
x9
4 18 x
x2
1
1
x8 8
30. lim
x→0
x
28. lim
x→2
1
1
2x 2
32. lim
x→0
x
csc x
34. lim
x→ cot x
36. lim
x→ 2
cos x 1
sin x
Section 12.2
In Exercises 37– 48, use a graphing utility to graph the
function and approximate the limit accurate to three
decimal places.
e2x 1
37. lim
x→0
x
1 ex
38. lim
x→0
x
39. lim x ln x
40. lim x2 ln x
sin 2x
41. lim
x→0
x
42. lim
sin 3x
x
44. lim
1 cos 2x
x
x→0
43. lim
x→0
x→0
x→0
tan x
x
x→ 0
3 x
1
x→1 1 x
47. lim 1 x2x
45. lim
x→0
46. lim
3 x x
x→ 1
x1
48. lim 1 2x1x
x→ 0
GRAPHICAL, NUMERICAL, AND ALGEBRAIC ANALYSIS
In Exercises 49–54, (a) graphically approximate the limit (if it
exists) by using a graphing utility to graph the function,
(b) numerically approximate the limit (if it exists) by using
the table feature of a graphing utility to create a table, and
(c) algebraically evaluate the limit (if it exists) by the
appropriate technique(s).
49. lim
x→1
51. lim
x1
x2 1
x→2 x 4
x4 1
3x2 4
4 x
53. lim x→16 x 16
50. lim
5x
25 x2
52. lim
x 4 2x2 8
x 4 6x2 8
x→5
x→2
54. lim
x→0
x
x 6
x6
x2
56. lim
x→2 x 2
x→6
1
x2 1
1
58. lim 2
x→1 x 1
x→1
2x 1,
60. lim f x where f x 4x,
4x ,
61. lim f x where f x 3 x,
4x,
62. lim f x where f x x 4,
x 1, x 2
59. lim f x where f x x→2
2x 3, x > 2
2
2
x→1
2
x→0
x→ 0
63. f x x cos x
64. f x x sin x
65. f x x sin x
66. f x x cos x
1
67. f x x sin
x
1
68. f x x cos
x
In Exercises 69 and 70, state which limit can be evaluated by
using direct substitution. Then evaluate or approximate each
limit.
69. (a) lim x 2 sin x 2
x→ 0
sin x 2
x2
x
70. (a) lim
x→ 0 cos x
(b) lim
x→ 0
(b) lim
x→ 0
1 cos x
x
In Exercises 71– 78, find lim
h→ 0
f x " h!
h
f x!
.
71. f x 2x 1
72. f x 3 4x
73. f x x
74. f x x 2
75. f x x 2 3x
76. f x 4 2x x 2
57. lim
x→1
869
In Exercises 63–68, use a graphing utility to graph the function
and the equations y ! x and y ! x in the same viewing
window. Use the graph to find lim f x!.
x 2 2
In Exercises 55– 62, graph the function. Determine the limit
(if it exists) by evaluating the corresponding one-sided limits.
55. lim
Techniques for Evaluating Limits
77. f x 1
x2
78. f x 1
x1
FREE-FALLING OBJECT In Exercises 79 and 80, use the
position function
s t! !
16t 2 " 256
x < 1
x 1
x 1
x > 1
which gives the height (in feet) of a free-falling object.
The velocity at time t ! a seconds is given by
lim [s a! s t!]/ a t!.
x 0
x > 0
80. Find the velocity when t 2 seconds.
t→a
79. Find the velocity when t 1 second.
870
Chapter 12
Limits and an Introduction to Calculus
81. SALARY CONTRACT A union contract guarantees an
8% salary increase yearly for 3 years. For a current
salary of $30,000, the salaries f t (in thousands of
dollars) for the next 3 years are given by
30.000, 0 < t
f t 32.400, 1 < t
34.992, 2 < t
1
2
3
x→5.5
5.3
5.4
5.5
5.6
5.7
6
?
C
C
4.5
4.9
(iii) lim Cx
x→500
EXPLORATION
TRUE OR FALSE? In Exercises 85 and 86, determine
whether the statement is true or false. Justify your answer.
86. If f c L, then lim f x L.
x→c
87. THINK ABOUT IT
(a) Sketch the graph of a function for which f 2 is
defined but for which the limit of f x as x
approaches 2 does not exist.
(b) Sketch the graph of a function for which the limit of
f x as x approaches 1 is 4 but for which f 1 4.
88. CAPSTONE
f x Given
2x,
x 1,
2
5
5.1
5.5
6
0.15x, 0 < x 25
0.10x, 25 < x 100
.
Cx 0.07x, 100 < x 500
0.05x, x > 500
(b) lim f x
x→0
(c) lim f x
x→0
89. WRITING Consider the limit of the rational function
given by pxqx. What conclusion can you make if
direct substitution produces each expression? Write a
short paragraph explaining your reasoning.
?
84. CONSUMER AWARENESS The cost C (in dollars) of
making x photocopies at a copy shop is given by the
function
x 0
,
x > 0
find each of the following limits. If the limit does not
exist, explain why.
(a) lim
px 0
qx 1
(b) lim
px 1
qx 1
(c) lim
px 1
qx 0
(d) lim
px 0
qx 0
x→c
x→100
(d) Explain how you can use the graph in part (a) to
verify that the limits in part (c) do not exist.
(a) lim f x
(c) Complete the table and observe the behavior of C as
x approaches 5. Does the limit of Cx as x
approaches 5 exist? Explain.
4
x→305
(ii) lim Cx
x→25
x→0
x
(iii) lim Cx
x→99
85. When your attempt to find the limit of a rational function
0
yields the indeterminate form 0, the rational function’s
numerator and denominator have a common factor.
(a) Use a graphing utility to graph C for 0 < x 10.
(b) Complete the table and observe the behavior of C as
x approaches 5.5. Use the graph from part (a) and
the table to find lim Cx.
5
(ii) lim Cx
(i) lim Cx
where x represents the weight of the package (in
pounds). Show that the limit of f as x → 1 does not exist.
83. CONSUMER AWARENESS The cost of hooking up
and towing a car is $85 for the first mile and $5 for each
additional mile or portion of a mile. A model for the
cost C (in dollars) is Cx 85 5 x 1, where
x is the distance in miles. (Recall from Section 1.6 that
f x x the greatest integer less than or equal to x.)
x
(i) lim Cx
(c) Create a table of values to show numerically that
each limit does not exist.
where t represents the time in years. Show that the limit
of f as t → 2 does not exist.
82. CONSUMER AWARENESS The cost of sending a
package overnight is $15 for the first pound and $1.30
for each additional pound or portion of a pound. A plastic
mailing bag can hold up to 3 pounds. The cost f x of
sending a package in a plastic mailing bag is given by
(b) Find each limit and interpret your result in the
context of the situation.
x→15
1
2
3
15.00, 0 < x
f x 16.30, 1 < x
17.60, 2 < x
(a) Sketch a graph of the function.
x→c
x→c
x→c
Section 12.3
871
The Tangent Line Problem
12.3 THE TANGENT LINE PROBLEM
What you should learn
• Use a tangent line to approximate
the slope of a graph at a point.
• Use the limit definition of slope
to find exact slopes of graphs.
• Find derivatives of functions and
use derivatives to find slopes of
graphs.
Why you should learn it
The slope of the graph of a function
can be used to analyze rates of change
at particular points on the graph. For
instance, in Exercise 74 on page 880,
the slope of the graph is used to
analyze the rate of change in book
sales for particular selling prices.
Tangent Line to a Graph
Calculus is a branch of mathematics that studies rates of change of functions. If you
go on to take a course in calculus, you will learn that rates of change have many
applications in real life.
Earlier in the text, you learned how the slope of a line indicates the rate at which a
line rises or falls. For a line, this rate (or slope) is the same at every point on the line.
For graphs other than lines, the rate at which the graph rises or falls changes from point
to point. For instance, in Figure 12.20, the parabola is rising more quickly at the point
x1, y1 than it is at the point x2, y2. At the vertex x3, y3, the graph levels off, and at
the point x4, y4, the graph is falling.
y
(x3, y3)
(x2, y2)
(x4, y4)
x
(x1, y1)
Bob Rowan, Progressive Image/Corbis
FIGURE
12.20
To determine the rate at which a graph rises or falls at a single point, you can find
the slope of the tangent line at that point. In simple terms, the tangent line to the graph
of a function f at a point Px1, y1 is the line that best approximates the slope of the
graph at the point. Figure 12.21 shows other examples of tangent lines.
y
y
y
P
P
P
x
FIGURE
x
x
12.21
From geometry, you know that a line is tangent to a circle if the line intersects the
circle at only one point. Tangent lines to noncircular graphs, however, can intersect
the graph at more than one point. For instance, in the first graph in Figure 12.21, if the
tangent line were extended, it would intersect the graph at a point other than the point
of tangency.
872
Chapter 12
Limits and an Introduction to Calculus
Slope of a Graph
Because a tangent line approximates the slope of the graph at a point, the problem of
finding the slope of a graph at a point is the same as finding the slope of the tangent line
at the point.
Example 1
y
Use the graph in Figure 12.22 to approximate the slope of the graph of f x x 2 at the
point 1, 1.
f(x) = x 2
5
Solution
4
From the graph of f x x 2, you can see that the tangent line at 1, 1 rises
approximately two units for each unit change in x. So, you can estimate the slope of the
tangent line at 1, 1 to be
3
2
2
1
1
Slope x
−3
−2
−1
1
2
change in y
change in x
3
−1
FIGURE
Visually Approximating the Slope of a Graph
"
12.22
2
1
2.
Because the tangent line at the point 1, 1 has a slope of about 2, you can
conclude that the graph of f has a slope of about 2 at the point 1, 1.
Now try Exercise 5.
When you are visually approximating the slope of a graph, remember that the
scales on the horizontal and vertical axes may differ. When this happens (as it
frequently does in applications), the slope of the tangent line is distorted, and you must
be careful to account for the difference in scales.
Example 2
Figure 12.23 graphically depicts the monthly normal temperatures (in degrees
Fahrenheit) for Dallas, Texas. Approximate the slope of this graph at the indicated point
and give a physical interpretation of the result. (Source: National Climatic Data Center)
Monthly Normal
Temperatures
y
Solution
2
90
Temperature (°F)
80
From the graph, you can see that the tangent line at the given point falls approximately
16 units for each two-unit change in x. So, you can estimate the slope at the given point to be
16
70
(10, 69)
Slope 60
50
"
40
30
change in y
change in x
16
2
8 degrees per month.
x
2
4
6
Month
FIGURE
Approximating the Slope of a Graph
12.23
8
10
12
This means that you can expect the monthly normal temperature in November to be
about 8 degrees lower than the normal temperature in October.
Now try Exercise 7.
Section 12.3
The Tangent Line Problem
873
Slope and the Limit Process
y
In Examples 1 and 2, you approximated the slope of a graph at a point by creating a
graph and then “eyeballing” the tangent line at the point of tangency. A more precise
method of approximating tangent lines makes use of a secant line through the point of
tangency and a second point on the graph, as shown in Figure 12.24. If x, f x is the
point of tangency and x h, f x h is a second point on the graph of f, the slope of
the secant line through the two points is given by
(x + h, f (x + h))
f (x + h ) − f (x )
msec (x, f (x))
h
FIGURE
x
12.24
y
Slope of secant line
The right side of this equation is called the difference quotient. The denominator h is the
change in x, and the numerator is the change in y. The beauty of this procedure is that you
obtain more and more accurate approximations of the slope of the tangent line by choosing
points closer and closer to the point of tangency, as shown in Figure 12.25.
y
(x + h, f (x + h))
change in y
f x h f x
.
change in x
h
y
(x + h, f (x + h))
y
(x + h, f (x + h))
(x, f (x))
(x, f (x))
f (x + h ) − f (x )
(x, f (x))
f (x + h ) − f (x )
Tangent line
f (x + h ) − f (x )
h
x
h
x
h
(x, f (x))
x
x
As h approaches 0, the secant line approaches the tangent line.
FIGURE 12.25
Using the limit process, you can find the exact slope of the tangent line
at x, f x.
Definition of the Slope of a Graph
The slope m of the graph of f at the point x, f x is equal to the slope of its
tangent line at x, f x, and is given by
m lim msec
h→ 0
lim
h→ 0
f x h f x
h
provided this limit exists.
From the definition above and from Section 12.2, you can see that the difference
quotient is used frequently in calculus. Using the difference quotient to find the slope
of a tangent line to a graph is a major concept of calculus.
874
Chapter 12
Limits and an Introduction to Calculus
Example 3
Finding the Slope of a Graph
Find the slope of the graph of f x x 2 at the point 2, 4.
Solution
Find an expression that represents the slope of a secant line at 2, 4.
y
msec f 2 h f 2
h
Set up difference quotient.
2 h2 22
h
Substitute in f x x2.
4 4h h 2 4
h
Expand terms.
4h h 2
h
Simplify.
h4 h
h
Factor and divide out.
f(x) = x 2
5
Tangent
line at
(− 2, 4)
4
3
4 h, h 0
2
Next, take the limit of msec as h approaches 0.
1
m = −4
x
−4
−3
Simplify.
1
−2
m lim msec lim 4 h 4
h→ 0
2
h→ 0
The graph has a slope of 4 at the point 2, 4, as shown in Figure 12.26.
FIGURE
12.26
Now try Exercise 9.
Example 4
Finding the Slope of a Graph
Find the slope of f x 2x 4.
Solution
f x h f x
h
Set up difference quotient.
lim
2x h 4 2x 4
h
Substitute in f x 2x 4.
lim
2x 2h 4 2x 4
h
Expand terms.
lim
2h
h
Divide out.
m lim
h→ 0
y
h→0
f(x) = −2x + 4
h→ 0
4
3
m = −2
h→ 0
2
2
1
x
−2
−1
1
−1
FIGURE
12.27
2
3
4
Simplify.
You know from your study of linear functions that the line given by
f x 2x 4 has a slope of 2, as shown in Figure 12.27. This conclusion is
consistent with that obtained by the limit definition of slope, as shown above.
Now try Exercise 11.
Section 12.3
The Tangent Line Problem
875
It is important that you see the difference between the ways the difference
quotients were set up in Examples 3 and 4. In Example 3, you were finding the slope
of a graph at a specific point c, f c. To find the slope in such a case, you can use the
following form of the difference quotient.
m lim
h→ 0
f c h f c
h
Slope at specific point
In Example 4, however, you were finding a formula for the slope at any point on the
graph. In such cases, you should use x, rather than c, in the difference quotient.
m lim
h→ 0
Try verifying the result in
Example 5 by using a graphing
utility to graph the function and
the tangent lines at 1, 2! and
2, 5! as
Example 5
Finding a Formula for the Slope of a Graph
Find a formula for the slope of the graph of f x x 2 1. What are the slopes at the
points 1, 2 and 2, 5?
Solution
y1 ! x2 " 1
2x
y3 ! 4x
f x h f x
h
Set up difference quotient.
x h2 1 x2 1
h
Substitute in f x x2 1.
x 2 2xh h 2 1 x 2 1
h
Expand terms.
2xh h 2
h
Simplify.
h2x h
h
Factor and divide out.
msec 3
in the same viewing window.
Some graphing utilities even
have a tangent feature that
automatically graphs the tangent
line to a curve at a given point. If
you have such a graphing utility,
try verifying Example 5 using
this feature.
2x h, h 0
y
f(x) = x 2 + 1
m lim msec lim 2x h 2x
h→ 0
6
Tangent
line at
(2, 5)
3
m 21 2
2
and at 2, 5, the slope is
x
−4 −3 −2 −1
−1
FIGURE
12.28
h→ 0
1
2
3
4
Formula for finding slope
Using the formula m 2x for the slope at x, f x, you can find the slope at the
specified points. At 1, 2, the slope is
5
4
Simplify.
Next, take the limit of msec as h approaches 0.
7
Tangent
line at
(−1, 2)
Formula for slope
Except for linear functions, this form will always produce a function of x, which can
then be evaluated to find the slope at any desired point.
T E C H N O LO G Y
y2 !
f x h f x
h
m 22 4.
The graph of f is shown in Figure 12.28.
Now try Exercise 17.
876
Chapter 12
Limits and an Introduction to Calculus
The Derivative of a Function
In Example 5, you started with the function f x x 2 1 and used the limit process
to derive another function, m 2x, that represents the slope of the graph of f at the
point x, f x. This derived function is called the derivative of f at x. It is denoted by
f ! x, which is read as “f prime of x.”
Definition of Derivative
In Section 1.5, you studied the
slope of a line, which represents
the average rate of change over
an interval. The derivative of a
function is a formula which
represents the instantaneous
rate of change at a point.
The derivative of f at x is given by
f ! x lim
h→ 0
f x h f x
h
provided this limit exists.
Remember that the derivative f ! x is a formula for the slope of the tangent line to the
graph of f at the point x, f x.
Example 6
Finding a Derivative
Find the derivative of f x 3x 2 2x.
Solution
f ! x lim
h→ 0
f x h f x
h
lim
3x h2 2x h 3x2 2x
h
lim
3x 2 6xh 3h 2 2x 2h 3x 2 2x
h
lim
6xh 3h 2 2h
h
lim
h6x 3h 2
h
h→0
h→ 0
h→ 0
h→0
lim 6x 3h 2
h→ 0
6x 2
So, the derivative of f x 3x 2 2x is
f ! x 6x 2.
Now try Exercise 33.
Note that in addition to f!x, other notations can be used to denote the derivative
of y f x. The most common are
dy
,
dx
y!,
d
f x,
dx
and
Dx y.
Section 12.3
Example 7
The Tangent Line Problem
877
Using the Derivative
Find f ! x for f x x. Then find the slopes of the graph of f at the points 1, 1 and
4, 2.
Solution
f ! x lim
h→ 0
Remember that in order to
rationalize the numerator of an
expression, you must multiply
the numerator and denominator
by the conjugate of the
numerator.
lim
lim
x h x
x h x
x h x
h
hx h x 1
x h x
1
2x
At the point 1, 1, the slope is
(4, 2)
(1, 1)
m=
1
f(x) =
−2
FIGURE
h
lim
lim
3
−1
x h x
hx h x h→ 0
−1
h→ 0
h→0
2
h
Because direct substitution yields the indeterminate form 00, you should use the rationalizing technique discussed in Section 12.2 to find the limit.
h→ 0
4
x h x
h→0
lim
y
f x h f x
h
m=
f!1 1
4
1
2
At the point 4, 2, the slope is
x
2
3
x
4
5
1
1
.
21 2
f ! 4 1
1
.
24 4
The graph of f is shown in Figure 12.29.
12.29
Now try Exercise 43.
CLASSROOM DISCUSSION
Using a Derivative to Find Slope In many applications, it is convenient to use
a variable other than x as the independent variable. Complete the following limit
process to find the derivative of f t! ! 3/t. Then use the result to find the slope
of the graph of f t! ! 3/t at the point 3, 1!.
f t " h!
f! t! ! lim
h→0
h
f t!
3
t"h
lim
h→0
h
3
t
!. . .
Write a short paragraph summarizing your findings.
878
Chapter 12
Limits and an Introduction to Calculus
EXERCISES
12.3
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1. _______ is the study of the rates of change of functions.
2. The _______ _______ to the graph of a function at a point is the line that best approximates the slope
of the graph at the point.
3. A _______ _______ is a line through the point of tangency and a second point on the graph.
4. The slope of the tangent line to a graph at x, f x is given by _______ .
SKILLS AND APPLICATIONS
In Exercises 5–8, use the figure to approximate the slope of
the curve at the point x, y!.
y
5.
y
6.
3
3
(x, y)
2
(x, y)
1
x
1
−1
2
x
4
−2 −1
1
3
−2
y
7.
8.
2
3
1
(x, y)
1
x
− 2 −1
1
2
1
−2 −1
2
9. gx x 2 4x, 3, 3
10. f x 10x 2x 2, 3, 12
11. gx 5 2x, 1, 3
(b) 1, 3
4
x1
14. gx (b) 8, 2
1
,
x2
4, 2
1
1, 3
18. f x x3
(a) 1, 1
(b) 2, 8
24. f x x 2 2x 1
26. f x x 3
28. f x 3
2x
In Exercises 29 – 42, find the derivative of the function.
29. f x 5
31. gx 9 31x
33. f x 4 3x2
1
x2
37. f x x 11
In Exercises 17–22, find a formula for the slope of the graph
of f at the point x, f x!!.Then use it to find the slope at the
two given points.
17. f x 4 x 2
(a) 0, 4
(b) 10, 3
35. f x 12. hx 2x 5, 1, 3
16. hx x 10,
22. f x x 4
(a) 5, 1
27. f x 3
In Exercises 9–16, use the limit process to find the slope of
the graph of the function at the specified point. Use a graphing
utility to confirm your result.
4
13. gx , 2, 2
x
15. hx x, 9, 3
21. f x x 1
(a) 5, 2
−2
−2
1
x2
(a) 0, 21 (b) 1, 1
x
3
20. f x (a) 0, 41 (b) 2, 12 23. f x x 2 2
25. f x 2 x
2
(x, y)
1
x4
In Exercises 23–28, sketch a graph of the function and the
tangent line at the point 1, f 1!!. Use the graph to approximate the slope of the tangent line.
y
3
19. f x 30. f x 1
32. f x 5x 2
34. f x x 2 3x 4
36. f x 1
x3
38. f x x 8
39. f x 1
x6
40. f x 1
x5
41. f x 1
x 9
42. hs 1
s 1
In Exercises 43–50, (a) find the slope of the graph of f at the
given point, (b) use the result of part (a) to find an equation
of the tangent line to the graph at the point, and (c) graph the
function and the tangent line.
43. f x x2 1,
44. f x 4 x2,
45. f x x3 2x,
46. f x x3 x,
2, 3
1, 3
1, 1
2, 6
Section 12.3
47. f x x 1,
48. f x x 2,
49. f x 70. f x 1
, 4, 1
x5
1.5
1
ln x
,
x
f ! x 71. PATH OF A BALL
is modeled by
1 ln x
x2
The path of a ball thrown by a child
y x2 5x 2
In Exercises 51–54, use a graphing utility to graph f over the
interval [ 2, 2] and complete the table. Compare the value
of the first derivative with a visual approximation of the slope
of the graph.
2
0.5
0
0.5
1
1.5
2
f x
f! x
51. f x 21x 2
52. f x 14 x3
53. f x x 3
x2 4
54. f x x4
In Exercises 55–58, find an equation of the line that is
tangent to the graph of f and parallel to the given line.
where y is the height of the ball (in feet) and x is the
horizontal distance (in feet) from the point from which
the ball was thrown. Using your knowledge of the
slopes of tangent lines, show that the height of the ball
is increasing on the interval 0, 2 and decreasing on the
interval 3, 5. Explain your reasoning.
72. PROFIT The profit P (in hundreds of dollars) that a
company makes depends on the amount x (in hundreds
of dollars) the company spends on advertising. The
profit function is given by
Px 200 30x 0.5x2.
Using your knowledge of the slopes of tangent lines,
show that the profit is increasing on the interval 0, 20
and decreasing on the interval 40, 60.
73. The table shows the revenues y (in millions of dollars)
for eBay, Inc. from 2000 through 2007. (Source:
eBay, Inc.)
Function
1
55. f x 4 x2
56. f x x2 1
Line
xy0
2x y 0
Year
Revenue, y
57. f x 58. f x x2 x
6x y 4 0
x 2y 6 0
2000
2001
2002
2003
2004
2005
2006
2007
431.4
748.8
1214.1
2165.1
3271.3
4552.4
5969.7
7672.3
1
2x 3
In Exercises 59–62, find the derivative of f. Use the derivative
to determine any points on the graph of f at which the
tangent line is horizontal. Use a graphing utility to verify your
results.
59. f x x 2 4x 3
61. f x 3x3 9x
879
69. f x x ln x, f ! x ln x 1
3, 2
3, 1
1
, 4, 1
50. f x x3
x
The Tangent Line Problem
60. f x x2 6x 4
62. f x x3 3x
In Exercises 63–70, use the function and its derivative to
determine any points on the graph of f at which the tangent
line is horizontal. Use a graphing utility to verify your results.
63. f x x 4 2x2, f ! x 4x3 4x
64. f x 3x4 4x3, f ! x 12x3 12x2
65. f x 2 cos x x, f ! x 2 sin x 1,
over the interval 0, 2
66. f x x 2 sin x, f ! x 1 2 cos x,
over the interval 0, 2
67. f x x 2e x, f ! x x2e x 2xe x
68. f x xex, f ! x ex xex
(a) Use the regression feature of a graphing utility to
find a quadratic model for the data. Let x represent
the time in years, with x 0 corresponding to
2000.
(b) Use a graphing utility to graph the model found in
part (a). Estimate the slope of the graph when x 5
and give an interpretation of the result.
(c) Use a graphing utility to graph the tangent line to
the model when x 5. Compare the slope given by
the graphing utility with the estimate in part (b).
880
Chapter 12
Limits and an Introduction to Calculus
74. MARKET RESEARCH The data in the table show the
number N (in thousands) of books sold when the price
per book is p (in dollars).
y
(c)
y
(d)
5
4
3
3
2
1
x
Price, p
Number of books, N
$10
$15
$20
$25
$30
$35
900
630
396
227
102
36
1 2 3
x
−2 −1
77. f x x
79. f x x
(a) Use the regression feature of a graphing utility to
find a quadratic model for the data.
(b) Use a graphing utility to graph the model found in
part (a). Estimate the slopes of the graph when
p $15 and p $30.
(c) Use a graphing utility to graph the tangent lines to
the model when p $15 and p $30. Compare
the slopes given by the graphing utility with your
estimates in part (b).
(d) The slopes of the tangent lines at p $15 and
p $30 are not the same. Explain what this means
to the company selling the books.
EXPLORATION
TRUE OR FALSE? In Exercises 75 and 76, determine
whether the statement is true or false. Justify your answer.
75. The slope of the graph of y x2 is different at every
point on the graph of f.
76. A tangent line to a graph can intersect the graph only at
the point of tangency.
In Exercises 77– 80, match the function with the graph of its
derivative. It is not necessary to find the derivative of the
function. [The graphs are labeled (a), (b), (c), and (d).]
y
(a)
1
x
−2
y
(b)
2 3
−2
−3
1
x
80. f x x 3
78. f x 81. THINK ABOUT IT Sketch the graph of a function
whose derivative is always positive.
82. THINK ABOUT IT Sketch the graph of a function
whose derivative is always negative.
83. THINK ABOUT IT Sketch the graph of a function for
which f!x < 0 for x < 1, f!x 0 for x > 1, and
f!1 0.
84. CONJECTURE Consider the functions f x x2 and
gx x3.
(a) Sketch the graphs of f and f! on the same set of
coordinate axes.
(b) Sketch the graphs of g and g! on the same set of
coordinate axes.
(c) Identify any pattern between the functions f and g
and their respective derivatives. Use the pattern to
make a conjecture about h!x if hx xn, where n
is an integer and n 2.
85. Consider the function f x 3x2 2x.
(a) Use a graphing utility to graph the function.
(b) Use the trace feature to approximate the coordinates
of the vertex of this parabola.
(c) Use the derivative of f x 3x2 2x to find the
slope of the tangent line at the vertex.
(d) Make a conjecture about the slope of the tangent
line at the vertex of an arbitrary parabola.
86. CAPSTONE Explain how the slope of the secant line
is used to derive the slope of the tangent line and the
definition of the derivative of a function f at a point
x, f x. Include diagrams or sketches as necessary.
5
4
3
2
1
x
−1
1 2 3
1 2 3 4 5
PROJECT: ADVERTISING To work an extended application
analyzing the amount spent on advertising in the United
States, visit this text’s website at academic.cengage.com.
(Data Source: Universal McCann)
Section 12.4
Limits at Infinity and Limits of Sequences
881
12.4 LIMITS AT INFINITY AND LIMITS OF SEQUENCES
What you should learn
• Evaluate limits of functions at
infinity.
• Find limits of sequences.
Why you should learn it
Finding limits at infinity is useful in
many types of real-life applications.
For instance, in Exercise 58 on page
889, you are asked to find a limit at
infinity to determine the number of
military reserve personnel in the
future.
Limits at Infinity and Horizontal Asymptotes
As pointed out at the beginning of this chapter, there are two basic problems in calculus:
finding tangent lines and finding the area of a region. In Section 12.3, you saw how
limits can be used to solve the tangent line problem. In this section and the next, you
will see how a different type of limit, a limit at infinity, can be used to solve the area
problem. To get an idea of what is meant by a limit at infinity, consider the function
given by
f x x1
.
2x
The graph of f is shown in Figure 12.30. From earlier work, you know that y 21 is a
horizontal asymptote of the graph of this function. Using limit notation, this can be
written as follows.
lim f x x→
© Karen Kasmauski/Corbis
lim f x x→ 1
2
1
2
Horizontal asymptote to the left
Horizontal asymptote to the right
These limits mean that the value of f x gets arbitrarily close to
increases without bound.
1
2
as x decreases or
y
3
2
y=1
2
f(x) = x + 1
2x
1
x
−3
−2
−1
x1
is
2x
a rational function. You can
review rational functions in
Section 2.6.
1
2
3
The function f x −2
−3
FIGURE
12.30
Definition of Limits at Infinity
If f is a function and L1 and L 2 are real numbers, the statements
lim f x L1
Limit as x approaches lim f x L 2
Limit as x approaches x→
and
x→ denote the limits at infinity. The first statement is read “the limit of f x as x
approaches is L1,” and the second is read “the limit of f x as x approaches
is L 2.”
882
Chapter 12
Limits and an Introduction to Calculus
To help evaluate limits at infinity, you can use the following definition.
Limits at Infinity
If r is a positive real number, then
lim
x→ 1
0.
xr
Limit toward the right
Furthermore, if xr is defined when x < 0, then
lim
x→
1
0.
xr
Limit toward the left
Limits at infinity share many of the properties of limits listed in Section 12.1. Some
of these properties are demonstrated in the next example.
Example 1
Evaluating a Limit at Infinity
Find the limit.
lim 4 x→ 3
x2
Algebraic Solution
Graphical Solution
Use the properties of limits listed in Section 12.1.
Use a graphing utility to graph y 4 3x2. Then use the trace
feature to determine that as x gets larger and larger, y gets closer and
closer to 4, as shown in Figure 12.31. Note that the line y 4 is a
horizontal asymptote to the right.
lim 4 x→ 3
3
lim 4 lim 2
x→ x→ x
x2
lim 4 3 lim
x→ x→ 1
x2
5
y=4
4 30
y = 4 − 32
x
4
So, the limit of f x 4 120
−20
3
as x approaches is 4.
x2
−1
FIGURE
12.31
Now try Exercise 9.
In Figure 12.31, it appears that the line y 4 is also a horizontal asymptote to the
left. You can verify this by showing that
lim
x→
4 x3 4.
2
The graph of a rational function need not have a horizontal asymptote. If it does,
however, its left and right horizontal asymptotes must be the same.
When evaluating limits at infinity for more complicated rational functions, divide
the numerator and denominator by the highest-powered term in the denominator. This
enables you to evaluate each limit using the limits at infinity at the top of this page.
Section 12.4
Example 2
Limits at Infinity and Limits of Sequences
883
Comparing Limits at Infinity
Find the limit as x approaches for each function.
a. f x 2x 3
3x 2 1
2x 2 3
3x 2 1
b. f x c. f x 2x 3 3
3x 2 1
Solution
In each case, begin by dividing both the numerator and denominator by x 2, the
highest-powered term in the denominator.
2x 3
a. lim
lim
x→ 3x2 1
x→ 2
3
2
x
x
1
3 2
x
0 0
30
0
2x2 3
b. lim
lim
x→ 3x2 1
x→ 3
x2
1
3 2
x
2 2 0
30
2
3
2x3 3
c. lim
lim
x→ 3x2 1
x→ 2x 3
3
x2
1
x2
In this case, you can conclude that the limit does not exist because the numerator
decreases without bound as the denominator approaches 3.
Now try Exercise 19.
In Example 2, observe that when the degree of the numerator is less than the degree
of the denominator, as in part (a), the limit is 0. When the degrees of the numerator and
denominator are equal, as in part (b), the limit is the ratio of the coefficients of the
highest-powered terms. When the degree of the numerator is greater than the degree of
the denominator, as in part (c), the limit does not exist.
This result seems reasonable when you realize that for large values of x, the
highest-powered term of a polynomial is the most “influential” term. That is, a polynomial
tends to behave as its highest-powered term behaves as x approaches positive or
negative infinity.
884
Chapter 12
Limits and an Introduction to Calculus
Limits at Infinity for Rational Functions
Consider the rational function f x NxDx, where
Nx an xn . . . a0
Dx bm xm . . . b0.
and
The limit of f x as x approaches positive or negative infinity is as follows.
0, n < m
lim f x an
x→ ± , nm
bm
If n > m, the limit does not exist.
Example 3
Finding the Average Cost
You are manufacturing greeting cards that cost $0.50 per card to produce. Your initial
investment is $5000, which implies that the total cost C of producing x cards is given
by C 0.50x 5000. The average cost C per card is given by
C
C 0.50x 5000
.
x
x
Find the average cost per card when (a) x 1000, (b) x 10,000, and (c) x 100,000.
(d) What is the limit of C as x approaches infinity?
Solution
a. When x 1000, the average cost per card is
C
0.501000 5000
1000
x 1000
$5.50.
b. When x 10,000, the average cost per card is
Average Cost
C
C
Average cost per card
(in dollars)
6
x 10,000
$1.00.
5
c. When x 100,000, the average cost per card is
4
3
0.5010,000 5000
10,000
C=
C
C 0.50x + 5000
=
x
x
2
0.50100,000 5000
100,000
x 100,000
$0.55.
1
d. As x approaches infinity, the limit of C is
x
60,000
100,000
y = 0.5 20,000
Number of cards
As x → , the average cost per card
approaches $0.50.
FIGURE 12.32
lim
x→ 0.50x 5000
$0.50.
x
The graph of C is shown in Figure 12.32.
Now try Exercise 55.
x→
Section 12.4
Limits at Infinity and Limits of Sequences
885
Limits of Sequences
You can review sequences in
Sections 9.1– 9.3.
Limits of sequences have many of the same properties as limits of functions. For
instance, consider the sequence whose nth term is an 12n.
1 1 1 1 1
, , , , ,. . .
2 4 8 16 32
As n increases without bound, the terms of this sequence get closer and closer to 0, and
the sequence is said to converge to 0. Using limit notation, you can write
T E C H N O LO G Y
There are a number of ways to
use a graphing utility to generate
the terms of a sequence. For
instance, you can display the
first 10 terms of the sequence
an
1
2n
using the sequence feature or
the table feature.
lim
n→ 1
0.
2n
The following relationship shows how limits of functions of x can be used to evaluate
the limit of a sequence.
Limit of a Sequence
Let f be a function of a real variable such that
lim f x L.
x→ If an! is a sequence such that f n an for every positive integer n, then
lim an L.
n→ A sequence that does not converge is said to diverge. For instance, the terms of the
sequence 1, 1, 1, 1, 1, . . . oscillate between 1 and 1. Therefore, the sequence
diverges because it does not approach a unique number.
Example 4
Finding the Limit of a Sequence
Find the limit of each sequence. (Assume n begins with 1.)
a. an 2n 1
n4
b. bn 2n 1
n2 4
c. cn 2n2 1
4n2
Solution
a. lim
2n 1
2
n4
3 5 7 9 11 13
, , , , , ,. . . → 2
5 6 7 8 9 10
b. n→
lim
2n 1
0
n2 4
3 5 7 9 11 13
, , , , , ,. . . → 0
5 8 13 20 29 40
c. lim
2n2 1 1
4n2
2
3 9 19 33 51 73
1
, , , ,
,
,. . . →
4 16 36 64 100 144
2
n→ You can use the definition of
limits at infinity for rational
functions on page 884 to verify
the limits of the sequences in
Example 4.
n→ Now try Exercise 39.
886
Chapter 12
Limits and an Introduction to Calculus
In the next section, you will encounter limits of sequences such as that shown in
Example 5. A strategy for evaluating such limits is to begin by writing the nth term in
standard rational function form. Then you can determine the limit by comparing the
degrees of the numerator and denominator, as shown on page 884.
Example 5
Finding the Limit of a Sequence
Find the limit of the sequence whose nth term is
an 8 nn 12n 1
.
n3
6
Algebraic Solution
Numerical Solution
Begin by writing the nth term in standard rational function
form—as the ratio of two polynomials.
Construct a table that shows the value of an as n
becomes larger and larger, as shown below.
an 8 nn 12n 1
n3
6
8nn 12n 1
6n3
8n3
12n2
4n
3n3
Write original nth term.
n
an
1
8
10
3.08
100
2.707
1000
2.671
10,000
2.667
Multiply fractions.
Write in standard rational form.
From this form, you can see that the degree of the numerator is equal
to the degree of the denominator. So, the limit of the sequence is the
ratio of the coefficients of the highest-powered terms.
From the table, you can estimate that as n approaches
8
, an gets closer and closer to 2.667 " 3.
8n3 12n2 4n 8
n→ 3n3
3
lim
Now try Exercise 49.
CLASSROOM DISCUSSION
Comparing Rates of Convergence In the table in Example 5 above, the value of
8
an approaches its limit of 3 rather slowly. (The first term to be accurate to three
decimal places is a4801 y 2.667.) Each of the following sequences converges to 0.
Which converges the quickest? Which converges the slowest? Why? Write a short
paragraph discussing your conclusions.
a. an
1
n
b. bn
1
n2
d. dn
1
n!
e. hn
2n
n!
c. cn
1
2n
Section 12.4
EXERCISES
12.4
887
Limits at Infinity and Limits of Sequences
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
1. A ________ at ________ can be used to solve the area problem in calculus.
2. When evaluating limits at infinity for complicated rational functions, you can divide the numerator and
denominator by the ________ term in the denominator.
3. A sequence that has a limit is said to ________.
4. A sequence that does not have a limit is said to ________.
SKILLS AND APPLICATIONS
In Exercises 5–8, match the function with its graph, using
horizontal asymptotes as aids. [The graphs are labeled (a),
(b), (c), and (d).]
y
(a)
(b)
6
2
4
2
1
2
x
3
− 4 −2
2
−2
−4
−3
−6
y
(c)
4
6
20.
2
lim
x→
x2 3
2 x2
x 1
x
lim
x→
2
24.
2x2 6
x 12
lim
x→
12x x4 2
26. lim 7 4
x→
2x2
x 32
3t1 t 5t 2
x
3x
lim 2x 1 x 3 27. lim
6
2x2 5x 12
x→ 1 6x 8x2
x→ 5x3 1
x→ 10x 3x2 7
4y 4
y2 3
lim
22. lim
3
2
t→ 6
4t 2
23. lim
25.
y
(d)
18. y→
lim
2t 1
t→ 3t 2t 2
21.
x
−2 −1
−1
t2
t3
19. lim
y
3
17. t→
lim
2
28.
x
x
−4 − 2
−2
2
4
6
−4 −2
−2
−4
−4
−6
−6
4x 2
5. f x 2
x 1
7. f x 4 1
x2
2
x→ 2
4
6
x2
6. f x 2
x 1
8. f x x 1
x
In Exercises 29–34, use a graphing utility to graph the function
and verify that the horizontal asymptote corresponds to the
limit at infinity.
29. y 3x
1x
30. y x2
x2 4
31. y 2x
1 x2
32. y 2x 1
x2 1
33. y 1 In Exercises 9–28, find the limit (if it exists). If the limit does
not exist, explain why. Use a graphing utility to verify your
result graphically.
x3 1
1x
lim 1 x
9. lim
x→ 11.
13.
2
x→ lim
x→
4x 3
2x 1
3x2 4
15. lim
x→ 1 x2
3x4 5
1 5x
lim 1 4x 10. lim
x→ 12.
x→ 14. lim
x→ 1 2x
x2
3x2 1
16. lim
x→ 4x2 5
3
x2
34. y 2 1
x
NUMERICAL AND GRAPHICAL ANALYSIS In Exercises
35–38, (a) complete the table and numerically estimate the
limit as x approaches infinity, and (b) use a graphing utility to
graph the function and estimate the limit graphically.
x
100
101
102
103
f x
35. f x 36. f x 37. f x 38. f x x x 2 2
3x 9x 2 1
32x 4x 2 x 44x 16x 2 x 104
105
106
888
Chapter 12
Limits and an Introduction to Calculus
In Exercises 39–48, write the first five terms of the sequence
and find the limit of the sequence (if it exists). If the limit
does not exist, explain why. Assume n begins with 1.
(a) What is the limit of S as t approaches infinity?
(b) Use a graphing utility to graph the function and
verify the result of part (a).
39. an n1
n2 1
40. an 3n
n2 2
(c) Explain the meaning of the limit in the context of
the problem.
41. an n
2n 1
42. an 4n 1
n3
55. AVERAGE COST The cost function for a certain
model of personal digital assistant (PDA) is given by
C 13.50x 45,750, where C is measured in dollars
and x is the number of PDAs produced.
(a) Write a model for the average cost per unit produced.
n2
2n 3
n 1!
45. an n!
4n2 1
2n
3n 1!
46. an 3n 1!
43. an 47. an 44. an 1n
n
48. an (b) Find the average costs per unit when x 100 and
x 1000.
(c) Determine the limit of the average cost function as
x approaches infinity. Explain the meaning of the
limit in the context of the problem.
1n1
n2
In Exercises 49–52, find the limit of the sequence. Then verify
the limit numerically by using a graphing utility to complete
the table.
100
n
101
102
103
104
105
106
an
1
1 nn 1
n
n
n
2
4
4 nn 1
50. an n n
n
2
49. an (c) Determine the limit of the average cost function as
x approaches infinity. Explain the meaning of the
limit in the context of the problem.
51. an 16 nn 12n 1
n3
6
52. an nn 1
1 nn 1
4
2
n
n
2
56. AVERAGE COST The cost function for a company
to recycle x tons of material is given by
C 1.25x 10,500, where C is measured in dollars.
(a) Write a model for the average cost per ton of
material recycled.
(b) Find the average costs of recycling 100 tons of
material and 1000 tons of material.
2
57. DATA ANALYSIS: SOCIAL SECURITY The table
shows the average monthly Social Security benefits B
(in dollars) for retired workers aged 62 or over
from 2001 through 2007. (Source: U.S. Social Security
Administration)
53. OXYGEN LEVEL Suppose that f t measures the level
of oxygen in a pond, where f t 1 is the normal
(unpolluted) level and the time t is measured in weeks.
When t 0, organic waste is dumped into the pond,
and as the waste material oxidizes, the level of oxygen
in the pond is given by
f t t2 t 1
.
t2 1
(a) What is the limit of f as t approaches infinity?
(b) Use a graphing utility to graph the function and
verify the result of part (a).
(c) Explain the meaning of the limit in the context of
the problem.
54. TYPING SPEED The average typing speed S (in
words per minute) for a student after t weeks of lessons
is given by
S
100t 2
,
65 t 2
t > 0.
Year
Benefit, B
2001
2002
2003
2004
2005
2006
2007
874
895
922
955
1002
1044
1079
A model for the data is given by
B
867.3 707.56t
, 1 ! t ! 7
1.0 0.83t 0.030t 2
where t represents the year, with t 1 corresponding to
2001.
Section 12.4
889
Limits at Infinity and Limits of Sequences
(a) Use a graphing utility to create a scatter plot of the
data and graph the model in the same viewing
window. How well does the model fit the data?
63. THINK ABOUT IT Find the functions f and g such that
both f x and gx increase without bound as x
approaches c, but lim f x gx exists.
(b) Use the model to predict the average monthly benefit
in 2014.
(c) Discuss why this model should not be used for
long-term predictions of average monthly Social
Security benefits.
64. THINK ABOUT IT
function given by
58. DATA ANALYSIS: MILITARY The table shows the
numbers N (in thousands) of U.S. military reserve
personnel for the years 2001 through 2007. (Source:
U.S. Department of Defense)
Year
Number, N
2001
2002
2003
2004
2005
2006
2007
1249
1222
1189
1167
1136
1120
1110
A model for the data is given by
N
1287.9 61.53t
,
1.0 0.08t
x→c
f x In Exercises 65–68, create a scatter plot of the terms of the
sequence. Determine whether the sequence converges or
diverges. If it converges, estimate its limit.
65. an 4 3 31 1.5n
67. an 1 1.5
(b) Use the model to predict the number of military
reserve personnel in 2014.
(c) What is the limit of the function as t approaches
infinity? Explain the meaning of the limit in the
context of the problem. Do you think the limit is
realistic? Explain.
66. an 3 2 31 0.5n
68. an 1 0.5
2 n
3 n
69. Use a graphing utility to graph the two functions given by
y1 1
and y2 x
1
3 x
in the same viewing window. Why does y1 not appear to
the left of the y-axis? How does this relate to the statement at the top of page 882 about the infinite limit
lim
x→
(a) Use a graphing utility to create a scatter plot of
the data and graph the model in the same viewing
window. How well does the model fit the data?
.
How many horizontal asymptotes does the function
appear to have? What are the horizontal asymptotes?
1 ! t ! 7
where t represents the year, with t 1 corresponding to
2001.
x
x2 1
Use a graphing utility to graph the
1
?
xr
70. CAPSTONE
Use the graph to estimate (a) lim f x,
x→ (b) lim f x, and (c) the horizontal asymptote of the
x→
graph of f.
y
(i)
y
(ii)
6
f
6
f
4
4
2
2
x
EXPLORATION
TRUE OR FALSE? In Exercises 59–62, determine whether
the statement is true or false. Justify your answer.
59. Every rational function has a horizontal asymptote.
60. If f x increases without bound as x approaches c, then
the limit of f x exists.
61. If a sequence converges, then it has a limit.
62. When the degrees of the numerator and denominator of
a rational function are equal, the limit does not exist.
x
−4
−2
−2
2
2
4
−2
71. Use a graphing utility to complete the table below to
verify that lim 1x 0.
x→ x
100
101
102
103
1
x
1
Make a conjecture about lim .
x→0 x
104
105
890
Chapter 12
Limits and an Introduction to Calculus
12.5 THE AREA PROBLEM
What you should learn
• Find limits of summations.
• Use rectangles to approximate areas
of plane regions.
• Use limits of summations to find
areas of plane regions.
Limits of Summations
Earlier in the text, you used the concept of a limit to obtain a formula for the sum S of
an infinite geometric series
S a1 a1r a1r 2 . . . #a r
1
i1
i1
Why you should learn it
The limits of summations are useful in
determining areas of plane regions.
For instance, in Exercise 50 on page
897, you are asked to find the limit of
a summation to determine the area of
a parcel of land bounded by a stream
and two roads.
a1
,
1r
r
< 1.
Using limit notation, this sum can be written as
n
S lim
n→ #
a1r i1 lim
n→ i1
a11 r n
a1
.
1r
1r
lim r n 0 for r < 1
n→ The following summation formulas and properties are used to evaluate finite and
infinite summations.
Summation Formulas and Properties
n
n
# c cn, c is a constant.
2.
nn 1(2n 1
i2 3.
6
i1
4.
1.
i1
i1
n
5.
# a
i
± bi i1
© Adam Woolfitt/Corbis
n
i
i1
Example 1
#i
3
i1
n
#a ± #b
nn 1
2
n
#
n
#i n
i
6.
i1
n 2n 12
4
n
# ka k # a , k is a constant.
i
i1
i
i1
Evaluating a Summation
Evaluate the summation.
200
# i 1 2 3 4 . . . 200
i1
Solution
Using the second summation formula with n 200, you can write
n
Recall from Section 9.3 that
the sum of a finite geometric
sequence is given by
n
#a r
1
i1
11 rr .
#i i1
200
#i i1
n
i1
a1
Furthermore, if 0 < r < 1,
then r n → 0 as n → .
nn 1
2
200200 1
2
40,200
2
20,100.
Now try Exercise 5.
Section 12.5
Example 2
The Area Problem
891
Evaluating a Summation
T E C H N O LO G Y
Evaluate the summation
Some graphing utilities have
a sum sequence feature that
is useful for computing
summations. Consult the user’s
guide for your graphing utility
for the required keystrokes.
n
S
3
4
5
n2
i2
2 2 2. . .
2
n
n
n
n
n2
i1
#
for n 10, 100, 1000, and 10,000.
Solution
Begin by applying summation formulas and properties to simplify S. In the
second line of the solution, note that 1n 2 can be factored out of the sum because n is
considered to be constant. You could not factor i out of the summation because i is the
(variable) index of summation.
n
S
i2
2
i1 n
#
1
n2
1
n2
Write original form of summation.
n
# i 2
Factor constant 1n2 out of sum.
i1
# i # 2
n
n
i1
i1
Write as two sums.
1 nn 1
2n
n2
2
1 n 2 5n
n2
2
n5
2n
Apply Formulas 1 and 2.
Add fractions.
Simplify.
Now you can evaluate the sum by substituting the appropriate values of n, as shown in
the following table.
n
n
#
i1
i2 n5
n2
2n
10
100
1000
10,000
0.75
0.525
0.5025
0.50025
Now try Exercise 15.
In Example 2, note that the sum appears to approach a limit as n increases. To find
the limit of
n5
2n
as n approaches infinity, you can use the techniques from Section 12.4 to write
lim
n→ n5 1
.
2n
2
892
Chapter 12
Limits and an Introduction to Calculus
Be sure you notice the strategy used in Example 2. Rather than separately evaluating
the sums
10
100
i2
,
2
i1 n
1000
i2
,
2
i1 n
#
#
#
i1
i2
,
n2
10,000
#
i1
i2
n2
it was more efficient first to convert to rational form using the summation formulas and
properties listed on page 890.
n
S
i2 n5
2
2n
i1 n
#
Summation
form
Rational form
With this rational form, each sum can be evaluated by simply substituting appropriate
values of n.
Example 3
Finding the Limit of a Summation
Find the limit of Sn as n → .
Sn # 1 n n n
i
2
1
i1
Solution
Begin by rewriting the summation in rational form.
As you can see from Example 3,
there is a lot of algebra involved
in rewriting a summation in
rational form. You may want
to review simplifying rational
expressions if you are having
difficulty with this procedure.
(See Appendix A.4.)
Sn # 1 n n n
i
2
1
Write original form
of summation.
i1
#
n
i1
1
n3
1
3
n
1
Square 1 in
and write as a single
fraction.
2ni i 2
Factor constant 1n3
out of the sum.
n 2 2ni i 2
n2
n
# n
2
n
i1
# n # 2ni # i n
n
n
i1
i1
2
i1
2
nn 1
1 3
nn 12n 1
n 2n
n3
2
6
14n3 9n2 n
6n3
Write as three sums.
$
In this rational form, you can now find the limit as n → .
lim Sn lim
n→ n→ 14
6
7
3
14n3 9n2 n
6n3
Now try Exercise 17.
Use summation
formulas.
Simplify.
Section 12.5
y
The Area Problem
893
The Area Problem
f
You now have the tools needed to solve the second basic problem of calculus: the area
problem. The problem is to find the area of the region R bounded by the graph of a
nonnegative, continuous function f, the x-axis, and the vertical lines x a and x b,
as shown in Figure 12.33.
If the region R is a square, a triangle, a trapezoid, or a semicircle, you can find its
area by using a geometric formula. For more general regions, however, you must use a
different approach—one that involves the limit of a summation. The basic strategy is to
use a collection of rectangles of equal width that approximates the region R, as
illustrated in Example 4.
x
a
FIGURE
b
Example 4
12.33
y
Use the five rectangles in Figure 12.34 to approximate the area of the region bounded
by the graph of f x 6 x 2, the x-axis, and the lines x 0 and x 2.
f (x ) = 6 − x 2
Solution
5
Because the length of the interval along the x-axis is 2 and there are five rectangles, the
width of each rectangle is 52. The height of each rectangle can be obtained by evaluating
f at the right endpoint of each interval. The five intervals are as follows.
4
3
0, 5,
5, 5,
2
2
1
x
1
FIGURE
Approximating the Area of a Region
2
3
5, 5,
2 4
5, 5,
4 6
6 8
5, 5 8 10
2
Notice that the right endpoint of each interval is 5i for i 1, 2, 3, 4, 5. The sum of the
areas of the five rectangles is
12.34
Height Width
# # 5
f
i1
2i
5
5 5
2
2i
6
5
5
i1
2
5
2
2
# 6 254 # i 5
5
i1
i1
2
2
4
65 5
25
2
44
30 5
5
212
8.48.
25
"
55 110 1
6
So, you can approximate the area of R as 8.48 square units.
Now try Exercise 23.
By increasing the number of rectangles used in Example 4, you can obtain
closer and closer approximations of the area of the region. For instance, using
2
25 rectangles of width 25
each, you can approximate the area to be A " 9.17 square
units. The following table shows even better approximations.
n
Approximate area
5
25
100
1000
5000
8.48
9.17
9.29
9.33
9.33
894
Chapter 12
Limits and an Introduction to Calculus
Based on the procedure illustrated in Example 4, the exact area of a plane region
R is given by the limit of the sum of n rectangles as n approaches .
Area of a Plane Region
Let f be continuous and nonnegative on the interval a, b. The area A of the
region bounded by the graph of f, the x-axis, and the vertical lines x a and
x b is given by
# f a n
A lim
n→ i1
b ai
n
Height
Example 5
ba
.
n
Width
Finding the Area of a Region
Find the area of the region bounded by the graph of f x x 2 and the x-axis between
x 0 and x 1, as shown in Figure 12.35.
y
1
Solution
f (x ) = x 2
Begin by finding the dimensions of the rectangles.
Width:
ba 10 1
n
n
n
Height: f a x
1
FIGURE
12.35
b ai
1 0i
i
i2
f 0
f
2
n
n
n
n
Next, approximate the area as the sum of the areas of n rectangles.
A"
# f a n
i1
i2
ba
n
n
1
Summation form
2
i1
# n n
n
b ai
n
i2
#n
i1
3
1 n 2
i
n3 i1
1 nn 12n 1
n3
6
2n3 3n2 n
6n3
#
Rational form
Finally, find the exact area by taking the limit as n approaches .
2n3 3n2 n 1
n→ 6n3
3
A lim
Now try Exercise 37.
Section 12.5
Example 6
y
The Area Problem
Finding the Area of a Region
Find the area of the region bounded by the graph of f x 3x x2 and the
x-axis between x 1 and x 2, as shown in Figure 12.36.
f (x ) = 3 x − x 2
2
Solution
Begin by finding the dimensions of the rectangles.
ba 21 1
n
n
n
Width:
1
Height: f a x
1
b ai
i
f 1
n
n
2
3 1
FIGURE
895
12.36
i
i
1
n
n
2
3
3i
2i
i2
1 2
n
n
n
2
i2
i
2
n n
Next, approximate the area as the sum of the areas of n rectangles.
A"
# f a n
i1
i2
i
ba
n
1
n
1
2
i1
# 2 n n n
n
b ai
n
n
#
2
i1
1 n
1 n 2
i
i
n2i1
n3 i1
#
#
1
1 nn 1
1 nn 12n 1
2n 2
3
n
n
2
n
6
2
n2 n 2n3 3n2 n
2n2
6n3
2
1
1
1
1
1
2 2n 3 2n 6n2
1
13
2
6
6n
Finally, find the exact area by taking the limit as n approaches .
A lim
n→ 136 6n1 2
13
6
Now try Exercise 43.
896
Chapter 12
Limits and an Introduction to Calculus
EXERCISES
12.5
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
VOCABULARY: Fill in the blanks.
n
1.
# c _______, c is a constant.
i1
n
2.
n
# i _______
3.
i1
#i
3
_______
i1
4. The exact _______ of a plane region R is given by the limit of the sum of n rectangles as n approaches .
SKILLS AND APPLICATIONS
In Exercises 5–12, evaluate the sum using the summation
formulas and properties.
60
5.
45
#7
6.
i1
20
7.
#
i3
8.
# k
3
#
y
y
4
3
1
2
i2
1
2
10.
1
# 2k 1
k1
10
j 2 j
12.
j1
#
In Exercises 13–20, (a) rewrite the sum as a rational function
S%n&, (b) use S%n& to complete the table, and (c) find lim S%n&.
101
102
103
104
#
14.
20
i
2
i1 n
#
8
6
#
#
i
1
# 1 n n i1
n
2
i1
20.
4
x
4
# 3 2 n n n
18.
#
i1
i
1
4
2i
2
n
n
2i
n
In Exercises 21–24, approximate the area of the region using
the indicated number of rectangles of equal width.
21. f x x 4
22. f x 2 x2
8
2
12
x
−4
2
−4
27. f x 91x3
4
6
28. f x 3 41x3
y
y
5
3
4
2
2
1
1
y
y
50
y
2i 3
16.
n2
i1
#
4
26. f x 9 x 2
n
3
15.
1 i 2
3
n
i1
n
i2
2 1
17.
3
n
n
n
i1
3
10
n
i3
4
i1 n
n
19.
8
y
n
n
2
Approximate area
25. f x 31x 4
Sn
13.
4
n
n→!
100
1
2
In Exercises 25–28, complete the table showing the approximate
area of the region in the graph using n rectangles of equal
width.
j 3 3j 2
j1
n
x
x
i1
50
k1
25
11.
#
24. f x 12x 13
2
i1
30
i1
20
9.
#3
23. f x 41x3
x
1
6
5
1
x
−2 −1
1 2 3
x
−1
1
2
3
x
−2 −1
1
2
3
Section 12.5
In Exercises 29–36, complete the table using the function
f %x&, over the specified interval [a, b], to approximate the area
of the region bounded by the graph of y f %x&, the x-axis,
and the vertical lines x a and x b using the indicated
number of rectangles. Then find the exact area as n → !.
4
n
8
20
50
100
897
The Area Problem
50. CIVIL ENGINEERING The table shows the measurements (in feet) of a lot bounded by a stream and two
straight roads that meet at right angles (see figure).
x
0
50
100
150
200
250
300
y
450
362
305
268
245
156
0
Approximate area
y
Road
450
Function
29. f x 2x 5
30. f x 3x 1
31. f x 16 2x
32. f x 20 2x
33. f x 9 x2
Interval
0, 4
0, 4
1, 5
2, 6
0, 2
4, 6
1, 3
2, 2
34. f x x2 1
35. f x 21 x 4
36. f x 21 x 1
270
180
x
50 100 150 200 250 300
(a) Use the regression feature of a graphing utility to
find a model of the form y ax3 bx2 cx d.
0, 1
0, 2
0, 1
2, 5
1, 1
0, 1
1, 2
1, 4
0, 1
0, 2
1, 4
1, 1
41. f x 2 x 2
42. f x x 2 2
43. gx 8 x3
44. g x 64 x3
45. gx 2x x3
46. gx 4x x3
47. f x 41x 2 4x
48. f x x 2 x3
TRUE OR FALSE? In Exercises 51 and 52, determine
whether the statement is true or false. Justify your answer.
51. The sum of the first n positive integers is nn 12.
52. The exact area of a region is given by the limit of the
sum of n rectangles as n approaches 0.
53. THINK ABOUT IT Determine which value best
approximates the area of the region shown in the graph.
(Make your selection on the basis of the sketch of the
region and not by performing any calculations.)
(a) 2 (b) 1 (c) 4 (d) 6 (e) 9
y
3
49. CIVIL ENGINEERING The boundaries of a parcel of
land are two edges modeled by the coordinate axes and
a stream modeled by the equation
y 3.0 #
106
x3
0.002x 2
(b) Use a graphing utility to plot the data and graph the
model in the same viewing window.
(c) Use the model in part (a) to estimate the area of
the lot.
EXPLORATION
Interval
37. f x 4x 1
38. f x 3x 2
39. f x 2x 3
40. f x 3x 4
Road
90
In Exercises 37–48, use the limit process to find the area of
the region between the graph of the function and the
x-axis over the specified interval.
Function
Stream
360
2
1
1.05x 400.
Use a graphing utility to graph the equation. Find the
area of the property. Assume all distances are measured
in feet.
x
1
3
54. CAPSTONE Describe the process of finding the area
of a region bounded by the graph of a nonnegative,
continuous function f, the x-axis, and the vertical lines
x a and x b.
898
Chapter 12
Limits and an Introduction to Calculus
12 CHAPTER SUMMARY
What Did You Learn?
Explanation/Examples
Review
Exercises
Use the definition of limit to
estimate limits (p. 851).
If f x becomes arbitrarily close to a unique number L as x
approaches c from either side, the limit of f x as x
approaches c is L. This is written as lim f x L.
1–4
Conditions Under Which Limits Do Not Exist
The limit of f x as x → c does not exist if any of the
following conditions are true.
1. f x approaches a different number from the right side of
c than it approaches from the left side of c.
5–8
x→c
Determine whether limits of
functions exist (p. 853).
Section 12.1
2. f x increases or decreases without bound as x approaches c.
3. f x oscillates between two fixed values as x approaches c.
Use properties of limits and direct
substitution to evaluate limits
(p. 855).
Let b and c be real numbers and let n be a positive integer.
1. lim b b
2. lim x c
3. lim x n c n
x→c
x→c
9–24
x→c
n c,
n x 4. lim for n even and c > 0
x→c
Properties of Limits
Let b and c be real numbers, let n be a positive integer, and
let f and g be functions where
lim f x L
x→c
and
lim gx K.
x→c
1. lim bf x bL
2. lim f x ± gx L ± K
3. lim f xgx LK
4. lim
x→c
x→c
x→c
x→c
f x
L
, K0
gx K
5. lim f xn Ln
Section 12.2
x→c
Use the dividing out technique
to evaluate limits of functions
(p. 861).
When evaluating a limit of a rational function by direct
substitution, you may encounter the indeterminate form
00. In this case, factor and divide out any common factors,
then try direct substitution again. (See Examples 1 and 2.)
25–32
Use the rationalizing technique
to evaluate limits of functions
(p. 863).
The rationalizing technique involves rationalizing the numerator
of the function when finding a limit. (See Example 3.)
33–36
Approximate limits of functions
(p. 864).
The table feature or zoom and trace features of a graphing utility
can be used to approximate limits. (See Examples 4 and 5.)
37–44
Evaluate one-sided limits of
functions (p. 865).
Limit from left: lim f x L1 or f x → L1 as x → c
45–52
x→c
Limit from right: lim f x L2 or f x → L2 as
x→c
Evaluate limits of difference
quotients from calculus (p. 867).
x → c
For any x-value, the limit of a difference quotient is an
f x h f x
.
expression of the form lim
h→0
h
53–56
Chapter Summary
What Did You Learn?
Explanation/Examples
Use a tangent line to approximate
the slope of a graph at a point
(p. 871).
The tangent line to the graph of a function f at
a point Px1, y1 is the line that best approximates
the slope of the graph at the point.
899
Review
Exercises
y
57– 64
P
Section 12.3
x
Use the limit definition of slope
to find exact slopes of graphs
(p. 873).
Definition of the Slope of a Graph
The slope m of the graph of f at the point x, f x is equal
to the slope of its tangent line at x, f x and is given by
f x h f x
h
m lim msec lim
h→0
65– 68
h→0
provided this limit exists.
Find derivatives of functions and
use derivatives to find slopes of
graphs (p. 876).
The derivative of f at x is given by
f x lim
h→0
69–82
f x h f x
h
Section 12.4
provided this limit exists. The derivative f x is a formula
for the slope of the tangent line to the graph of f at the point
x, f x.
Evaluate limits of functions at
infinity (p. 881).
Find limits of sequences (p. 885).
If f is a function and L1 and L2 are real numbers, the statements
lim f x L1 and lim f x L2 denote the limits at infinity.
83–92
Limit of a Sequence
93–98
x→
x→ Let f be a function of a real variable such that lim f x L.
x→ If !an" is a sequence such that f n an for every positive
integer n, then lim an L.
n→ Find limits of summations
(p. 890).
n
1.
n
2.
c cn, c is a constant.
i1
i1
n
n
i2 5.
ai ± bi i1
nn 1
2
n
i3 4.
i1
n2n 12
4
n
ai ±
i1
n
6.
i
i1
nn 12n 1
6
n
3.
Section 12.5
99, 100
Summation Formulas and Properties
bi
i1
n
ai, k is a constant.
kai k
i1
i1
Use rectangles to approximate
areas of plane regions (p. 893).
A collection of rectangles of equal width can be used to
approximate the area of a region. Increasing the number of
rectangles gives a closer approximation. (See Example 4.)
101–104
Use limits of summations to find
areas of plane regions (p. 894).
Area of a Plane Region
Let f be continuous and nonnegative on a, b. The area A
of the region bounded by the graph of f, the x-axis, and the
vertical lines x a and x b is given by
105 –113
n
A lim
n→ i1
f a
b ai
n
b n a.
900
Chapter 12
Limits and an Introduction to Calculus
12 REVIEW EXERCISES
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
12.1 In Exercises 1–4, complete the table and use the
result to estimate the limit numerically. Determine whether
or not the limit can be reached.
In Exercises 9 and 10, use the given information to evaluate
each limit.
9. lim f x 4, lim gx 5
x→c
1. lim 6x 1
x→3
x→c
(a) lim f x3
(b) lim 3f x gx
(c) lim f xgx
(d) lim
x→c
2.9
x
2.99
2.999
3
3.001
3.01
3.1
x→c
f x
?
x→c
x→c
1.9
1.99
1.999
f x
2
2.001
2.01
2.1
?
(c) lim f xgx
x→c
1
x→4
13. lim
0.01
0.001
f x
0
0.001
0.01
0.1
?
x→2
x2
1
x3 2
f x
0
0.001
0.01
0.1
?
In Exercises 5–8, use the graph to find the limit (if it exists).
If the limit does not exist, explain why.
6. lim
x→1
x→2
y
1
x2
x
1 2 3
−2
x2 1
x→1 x 1
2 3 4 5
−1
−2
−3
8. lim 2x2 1
7. lim
x→1
y
4
3
2
1
24. lim arctan x
x→0
−1
−2
1 2 3
3x 5
5x 3
x→0
26. lim
27. lim
x5
x 2 5x 50
28. lim
x2 4
x→2 x3 8
1
1
x2
31. lim
x→1
x1
29. lim
x→1
− 2 −1
x→2
t2
t2 4
36. lim
x
x→2
25. lim
x→5
x
x→2
12.2 In Exercises 25–36, find the limit (if it exists). Use
a graphing utility to verify your result graphically.
35. lim
1 2 3 4
x→e
23. lim 2e x
u→ 0
5
4
3
14. lim 7
22. lim
33. lim
y
x→1
t2 1
21. lim
t→3
t
x→5
x
12. lim 5 x
3 4x
20. lim t→2
3
2
1
−1
x→c
19. lim 5x 33x 5
y
3
2
1
(d) lim f x 2gx
18. lim 5 2x x2
x→1
5. lim 3 x
f x
18
17. lim 5x 4
x→2
0.001
x→c
16. lim tan x
x→3
ln1 x
x→0
x
0.01
(b) lim
15. lim sin 3x
x→ 4. lim
0.1
x
11. lim 2 x 3
1 ex
x→0
x
x
3 f
In Exercises 11–24, find the limit by direct substitution.
3. lim
0.1
f x
gx
x→c
(a) lim
x→2
x
x→c
10. lim f x 27, lim gx 12
2. lim x2 3x 1
x
x→c
4 u 2
u
x 1 2
x5
3 x 2
1x
x→5
5x
x2 25
x→1
x1
x2 5x 6
t 3 27
t→3 t 3
1
1
x1
32. lim
x→ 0
x
30. lim
34. lim
v→0
v 9 3
v
901
Review Exercises
GRAPHICAL AND NUMERICAL ANALYSIS In Exercises
37–44, (a) graphically approximate the limit (if it exists)
by using a graphing utility to graph the function, and
(b) numerically approximate the limit (if it exists) by using
the table feature of a graphing utility to create a table.
37. lim
x3
x2 9
39. lim
e2x
x→3
x→0
40. lim
2
e4x
42. lim
tan 2x
x
x→0
x→0
2x 1 3
x1
x→1
4x
16 x2
x→4
sin 4x
41. lim
x→0
2x
43. lim
38. lim
44. lim
x →1
x 3
46. lim
x3
2
47. lim 2
x→2 x 4
x5
49. lim
x→5 x 5
1 x
x1
8 x
x 3,
x 6,
52. lim f x where f x x 4,
51. lim f x where f x 5 x,
x→2
2
x→0
2
In Exercises 53–56, find lim
f #x
h→0
53. f x 4x 3
55. f x 3x x 2
x ! 2
x > 2
x 0
x < 0
h$ ! f #x$
.
h
2
5
x
−1
−2
−3
−4
1
3
(x, y)
68. f x x
(a) 1, 1
(x, y)
5
In Exercises 59–64, sketch a graph of the function and
the tangent line at the point #2, f #2$$. Use the graph to
approximate the slope of the tangent line.
61. f x x 2
62. f x x2 5
1
80. f x 12 x
4x
2x 3
84. lim
7x
14x 2
85. lim
3x
3x
86. lim
1 2x
x2
x→ 60. f x 6 x2
1
x 4
83. lim
x
59. f x x 2 2x
6
5t
78. gt 81. f x 2x2 1, 0, 1
82. f x x2 10, 2, 14
x→ 1 2 3
70. gx 3
72. f x 3x
74. f x x3 4x
76. gt t 3
12.4 In Exercises 83–92, find the limit (if it exists). If the
limit does not exist, explain why. Use a graphing utility to
verify your result graphically.
3
2
1
−1
(b) 4, 2
In Exercises 81 and 82, (a) find the slope of the graph of f
at the given point, (b) use the result of part (a) to find an
equation of the tangent line to the graph at the point, and
(c) graph the function and the tangent line.
y
58.
4
x6
(a) 7, 4
(b) 8, 2
79. gx 54. f x 11 2x
56. f x x2 5x 2
y
In Exercises 65–68, find a formula for the slope of the graph
of f at the point #x, f #x$$. Then use it to find the slope at the
two given points.
69. f x 5
71. hx 5 12x
73. gx 2x2 1
75. f t t 5
4
77. gs s5
12.3 In Exercises 57 and 58, approximate the slope of the
tangent line to the graph at the point #x, y$.
57.
1
3x
In Exercises 69–80, find the derivative of the function.
x→8
64. f x 67. f x 8x
1
48. lim 2
x→3 x 9
x2
50. lim
x→2 x 2
x→3
6
x4
65. f x x 2 4x
(a) 0, 0
(b) 5, 5
66. f x 14 x4
(a) 2, 4 (b) 1, 41 In Exercises 45–52, graph the function. Determine the limit
(if it exists) by evaluating the corresponding one-sided limits.
45. lim
63. f x 87. lim
x→
x→ x→ 2x
x 2 25
89. lim
x2
2x 3
91. lim
x 2
x→ x→ 88.
lim
x→
3x
1 x3
3y 4
y→ y 1
90. lim
x
2
3
2
92. lim 2 x→ 2x2
x 12
902
Chapter 12
Limits and an Introduction to Calculus
In Exercises 93–98, write the first five terms of the sequence
and find the limit of the sequence (if it exists). If the limit
does not exist, explain why. Assume n begins with 1.
93. an 4n 1
3n 1
95. an 1
n3
97. an n2
3n 2
94. an Function
n
n2 1
1
n
1
98. an 2 3 2nn 1
2n
n
n1
96. an 12.5 In Exercises 99 and 100, (a) use the summation formulas
and properties to rewrite the sum as a rational function S#n$,
(b) use S#n$ to complete the table, and (c) find lim S#n$.
n→"
100
n
101
102
103
n
i1
n
4i 2
2
i
n
n
1
n
4 n n 2
3i
100.
i1
3i
2
In Exercises 101 and 102, approximate the area of the region
using the indicated number of rectangles of equal width.
101. f x 4 x
Interval
105. f x 10 x
106. f x 2x 6
107. f x x 2 4
108. f x 8x x 2
109. f x x 3 1
110. f x 1 x3
111. f x 2x 2 x3
112. f x 4 x 22
0, 10
3, 6
1, 2
0, 1
0, 2
3, 1
1, 1
0, 4
113. CIVIL ENGINEERING The table shows the measurements (in feet) of a lot bounded by a stream and two
straight roads that meet at right angles (see figure).
104
Sn
99.
In Exercises 105–112, use the limit process to find the area
of the region between the graph of the function and the
x-axis over the specified interval.
x
0
100
200
300
400
500
y
125
125
120
112
90
90
x
600
700
800
900
1000
y
95
88
75
35
0
102. f x 4 x2
y
y
y
Road
125
4
3
2
3
100
2
75
1
1
50
x
x
1 2
25
1
−1
3 4
Stream
Road
x
200 400 600 800 1000
In Exercises 103 and 104, complete the table to show the
approximate area of the region in the graph using n
rectangles of equal width.
4
n
8
20
50
(a) Use the regression feature of a graphing utility to
find a model of the form y ax3 bx2 cx d.
(b) Use a graphing utility to plot the data and graph
the model in the same viewing window.
(c) Use the model in part (a) to estimate the area of the lot.
Approximate area
EXPLORATION
103. f x 14x2
104. f x 4x x 2
y
TRUE OR FALSE? In Exercises 114 and 115, determine
whether the statement is true or false. Justify your answer.
y
4
4
3
2
3
2
1
1
x
1
2
3
4
x
1 2
3
114. The limit of the sum of two functions is the sum of the
limits of the two functions.
115. If the degree of the numerator Nx of a rational
function f x NxDx is greater than the degree
of its denominator Dx, then the limit of the rational
function as x approaches is 0.
116. WRITING Write a paragraph explaining several
reasons why the limit of a function may not exist.
903
Chapter Test
12 CHAPTER TEST
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Take this test as you would take a test in class. When you are finished, check your work
against the answers given in the back of the book.
In Exercises 1–3, sketch a graph of the function and approximate the limit (if it exists).
Then find the limit (if it exists) algebraically by using appropriate techniques.
1. lim
x→2
x2 1
2x
2. lim
x→1
x2 5x 3
1x
3. lim
x 2
x→5
x5
In Exercises 4 and 5, use a graphing utility to graph the function and approximate the
limit. Write an approximation that is accurate to four decimal places. Then create a table
to verify your limit numerically.
e2x 1
x→0
x
sin 3x
x→0
x
4. lim
5. lim
6. Find a formula for the slope of the graph of f at the point x, f x. Then use the
formula to find the slope at the given point.
(a) f x 3x2 5x 2, 2, 0
(b) f x 2x3 6x, 1, 8
In Exercises 7–9, find the derivative of the function.
2
7. f x 5 x
5
y
8. f x 2x2 4x 1
9. f x 1
x3
In Exercises 10–12, find the limit (if it exists). If the limit does not exist, explain why. Use
a graphing utility to verify your result graphically.
10
10. lim
x→ 6
4
6
5x 1
1 3x2
x→ x2 5
11. lim
12.
lim
x→
x2
3x 2
In Exercises 13 and 14, write the first five terms of the sequence and find the limit of the
sequence (if it exists). If the limit does not exist, explain why. Assume n begins with 1.
2
x
1
2
13. an −2
FIGURE FOR
14. an 1 1n
n
15. Approximate the area of the region bounded by the graph of f x 8 2x2 shown
at the left using the indicated number of rectangles of equal width.
15
Time
(seconds), x
Altitude
(feet), y
0
1
2
3
4
5
0
1
23
60
115
188
TABLE FOR
n2 3n 4
2n2 n 2
18
In Exercises 16 and 17, use the limit process to find the area of the region between the
graph of the function and the x-axis over the specified interval.
16. f x x 2; interval: 2, 2
17. f x 3 x2; interval: 1, 1
18. The table shows the altitude of a space shuttle during its first 5 seconds of motion.
(a) Use the regression feature of a graphing utility to find a quadratic model
y ax2 bx c for the data.
(b) The value of the derivative of the model is the rate of change of altitude with
respect to time, or the velocity, at that instant. Find the velocity of the shuttle
after 5 seconds.
904
Chapter 12
Limits and an Introduction to Calculus
www.CalcChat.com for worked-out
12 CUMULATIVE TEST FOR CHAPTERS 10 –12 See
solutions to odd-numbered exercises.
Take this test as you would take a test in class. When you are finished, check your work
against the answers given in the back of the book.
In Exercises 1 and 2, identify the conic and sketch its graph.
1.
x 22 y 12
1
4
9
2. x 2 y 2 2x 4y 1 0
3. Find the standard form of the equation of the ellipse with vertices 0, 0 and 0, 4
and endpoints of the minor axis 1, 2 and 1, 2.
4. Determine the number of degrees through which the axis must be rotated to
eliminate the xy-term of the conic x 2 4xy 2y 2 6. Then graph the conic.
5. Sketch the curve represented by the parametric equations x 4 ln t and y 21 t 2.
Then eliminate the parameter and write the corresponding rectangular equation
whose graph represents the curve.
6. Plot the point 2, 34 and find three additional polar representations for
2 < % < 2.
7. Convert the rectangular equation 8x 3y 5 0 to polar form.
8. Convert the polar equation r z
In Exercises 9–11, sketch the graph of the polar equation. Identify the type of graph.
4
(0, 4, 3)
(0, 0, 3)
2
2
6
10. r 3 2 sin %
11. r 2 5 cos %
4
y
12. The point is located six units behind the yz-plane, one unit to the right of the
xz-plane, and three units above the xy-plane.
13. The point is located on the y-axis, four units to the left of the xz-plane.
14. Find the distance between the points 2, 3, 6 and 4, 5, 1.
4
x
FIGURE FOR
9. r In Exercises 12 and 13, find the coordinates of the point.
(0, 0, 0)
2
2
to rectangular form.
4 5 cos %
15
15. Find the lengths of the sides of the right triangle at the left. Show that these lengths
satisfy the Pythagorean Theorem.
16. Find the coordinates of the midpoint of the line segment joining 3, 4, 1 and
5, 0, 2.
17. Find an equation of the sphere for which the endpoints of a diameter are 0, 0, 0
and 4, 4, 8.
18. Sketch the graph of the equation x 22 y 12 z2 4, and sketch the
xy-trace and the yz-trace.
19. For the vectors u %2, 6, 0& and v %4, 5, 3&, find u
# v and u " v.
In Exercises 20–22, determine whether u and v are orthogonal, parallel, or neither.
20. u %4, 4, 0&
v %0, 8, 6&
21. u %4, 2, 10&
v %2, 6, 2&
22. u %1, 6, 3&
v %3, 18, 9&
23. Find sets of (a) parametric equations and (b) symmetric equations for the line
passing through the points 2, 3, 0 and 5, 8, 25.
24. Find the parametric form of the equation of the line passing through the point
1, 2, 0 and perpendicular to 2x 4y z 8.
Cumulative Test for Chapters 10–12
z
6
(−1, −1, 3)
(0, 0, 0)
(3, −1, 3)
(−1, 3, 3)
−4
4
(2, 2, 0)
4
x
FIGURE FOR
28
25. Find an equation of the plane passing through the points 0, 0, 0, 2, 3, 0, and
5, 8, 25.
26. Sketch the graph and label the intercepts of the plane given by 3x 6y 12z 24.
27. Find the distance between the point 0, 0, 25 and the plane 2x 5y z 10.
(3, 3, 3)
(2, 0, 0)
905
y
(0, 2, 0)
28. A plastic wastebasket has the shape and dimensions shown in the figure. In
fabricating a mold for making the wastebasket, it is necessary to know the angle
between two adjacent sides. Find the angle.
In Exercises 29–34, find the limit (if it exists). If the limit does not exist, explain why.
Use a graphing utility to verify your result graphically.
29. lim
x 4 2
30. lim
x
x→ 0
x→4
1
1
x3 3
32. lim
x→0
x
33. lim
x 4
31. lim sin
x4
x→0
x 16 4
34. lim
x
x→0
x→2
x x2
x2 4
In Exercises 35–38, find a formula for the slope of the graph of f at the point #x, f #x$$.
Then use the formula to find the slope at the given point.
35. f x 4 x 2, 2, 0
1
1
1,
,
37. f x x3
4
36. f x x 3,
2, 1
38. f x x 2 x, 1, 0
In Exercises 39–44, find the limit (if it exists). If the limit does not exist, explain why.
Use a graphing utility to verify your result graphically.
39. lim
x→ 2x 4 x3 4
x2 9
40. lim
x3
x 9
41. lim
3 7x
x4
43. lim
2x
x2 3x 2
44. lim
3x
x2 1
x→ 3x2 1
x→ x2 4
42. lim
x→ 2
x→ x→ In Exercises 45–47, evaluate the sum using the summation formulas and properties.
50
20
1 i2
45.
46.
40
3k 2 2k
47.
k1
i1
12 i3
i1
In Exercises 48 and 49, approximate the area of the region using the indicated number of
rectangles of equal width.
y
48.
y
49.
7
6
5
4
3
2
1
2
y = 2x
y=
1
x2 + 1
x
1
2
x
3
−1
1
In Exercises 50–52, use the limit process to find the area of the region between the graph
of the function and the x-axis over the specified interval.
50. f x 1 x3
Interval: 0, 1
51. f x x 2
Interval: 0, 1
52. f x 4 x2
Interval: 0, 2
PROOFS IN MATHEMATICS
Many of the proofs of the definitions and properties presented in this chapter are
beyond the scope of this text. Included below are simple proofs for the limit of a power
function and the limit of a polynomial function.
Proving Limits
To prove most of the definitions
and properties in this chapter,
you must use the formal
definition of limit. This
definition is called the epsilondelta definition and was first
introduced by Karl Weierstrass
(1815–1897). If you go on to
take a course in calculus, you
will use this definition of limit
extensively.
Limit of a Power Function
n
(p. 855)
n
lim x c , c is a real number and n is a positive integer.
x→c
Proof
lim xn limx # x # x # . . .
x→c
x→c
# x
n factors
lim x # lim x # lim x # . . . # lim x
x→c
x→c
x→c
Product Property of Limits
x→c
n factors
c#c#c#. . .#c
Limit of the identity function
n factors
c
n
Exponential form
Limit of a Polynomial Function
(p. 857)
If p is a polynomial function and c is a real number, then
lim px pc.
x→c
Proof
Let p be a polynomial function such that
px an x n an1 x n1 . . . a2 x 2 a1x a0.
Because a polynomial function is the sum of monomial functions, you can write the
following.
lim px lim an x n an1 x n1 . . . a2 x 2 a1x a0
x→c
x→c
lim an x n lim an1x n1 . . . lim a2 x 2 lim a1x lim a0
x→c
906
x→c
x→c
x→c
x→c
ancn an1cn1 . . . a2c2 a1c a0
Scalar Multiple Property
of Limits and limit of a
power function
pc
p evaluated at c
PROBLEM SOLVING
This collection of thought-provoking and challenging exercises further explores and
expands upon concepts learned in this chapter.
1. Consider the graphs of the four functions g1, g2, g3, and g4.
y
y
3
3
g1
2
g2
2
x→3
your answer in part (b)?
5. Find the values of the constants a and b such that
1
1
x
x
1
2
1
3
2
(c) Let Qx, y be another point on the circle in the first
quadrant. Find the slope mx of the line joining P and
Q in terms of x.
(d) Evaluate lim mx . How does this number relate to
lim
3
a bx 3
x
x→0
y
3.
y
6. Consider the function given by
3
3
g3
2
g4
f x 2
1
1
2
x
3
1
2
(b) lim f x 3
x→2
x→2
(c) lim f x 3
x→2
2. Sketch the graph of the function
f x x x.
(a) Evaluate f 1, f 0, f 12 , and f 2.7.
given
x→12
(a) Evaluate f 4 , f 3, and f 1.
(b) Evaluate the following limits.
'1x(.
1
lim f x, lim f x,
x→1
lim
x→ 12
f x,
lim
x→ 12
2
2
f x
4. Let P3, 4 be a point on the circle x y 25 (see
figure).
y
6
P(3, 4)
2
−6
−2 O
Q
x
2
0,1,
if x is rational
if x is irrational
0,x,
if x is rational
.
if x is irrational
and
3. Sketch the graph of the function given by f x x→1
(d) Evaluate lim f x. Verify your result using the graph
x→1
in part (b).
7. Let
f x lim f x, lim f x, lim f x
x→1
(c) Evaluate lim f x. Verify your result using the
x→27
graph in part (b).
by
(b) Evaluate the following limits.
x→1
.
(b) Use a graphing utility to graph the function.
3
For each given condition of the function f, which of the
graphs could be the graph of f ?
(a) lim f x 3
x1
(a) Find the domain of f.
x
1
3 x13 2
6
−6
gx Find (if possible) lim f x and lim gx. Explain your
x→0
x→0
reasoning.
8. Graph the two parabolas y x 2 and y x2 2x 5
in the same coordinate plane. Find equations of the two
lines that are simultaneously tangent to both parabolas.
9. Find a function of the form f x a bx that is
tangent to the line 2y 3x 5 at the point 1, 4.
10. (a) Find an equation of the tangent line to the parabola
y x2 at the point 2, 4.
(b) Find an equation of the normal line to y x 2 at the
point 2, 4. (The normal line is perpendicular to
the tangent line.) Where does this line intersect the
parabola a second time?
(c) Find equations of the tangent line and normal line to
y x 2 at the point 0, 0.
(a) What is the slope of the line joining P and O0, 0?
(b) Find an equation of the tangent line to the circle at P.
907
11. A line with slope m passes through the point 0, 4.
(a) Recall that the distance d between a point x1, y1
and the line Ax By C 0 is given by
d
Ax1 By1 C
.
Write the distance d between the line and the point
3, 1 as a function of m.
(b) Use a graphing utility to graph the function from
part (a).
m→ lim
x→0
A2 B2
(c) Find lim dm and
13. When using a graphing utility to generate a table to
approximate
lim dm. Give a geometric
tan 2x
x
a student concluded that the limit was 0.03491 rather
than 2. Determine the probable cause of the error.
14. Let Px, y be a point on the parabola y x 2 in the first
quadrant. Consider the triangle PAO formed by P,
A0, 1, and the origin O0, 0, and the triangle PBO
formed by P, B1, 0, and the origin (see figure).
m→
y
interpretation of the results.
12. A heat probe is attached to the heat exchanger of a
heating system. The temperature T (in degrees Celsius)
is recorded t seconds after the furnace is started. The
results for the first 2 minutes are recorded in the table.
P
A
1
B
O
t
T
0
15
30
45
60
75
90
105
120
25.2&
36.9&
45.5&
51.4&
56.0&
59.6&
62.0&
64.0&
65.2&
x
1
(a) Write the perimeter of each triangle in terms of x.
(b) Complete the table. Let rx be the ratio of the
perimeters of the two triangles.
rx Perimeter
Perimeter
PAO
PBO
4
x
Perimeter
PAO
Perimeter
PBO
2
1
0.1
0.01
rx
(a) Use the regression feature of a graphing utility to
find a model of the form T1 at 2 bt c for the
data.
(b) Use a graphing utility to graph T1 with the original
data. How well does the model fit the data?
(c) A rational model for the data is given by
T2 (c) Find lim rx.
x→0
15. Archimedes showed that the area of a parabolic arch is
equal to 23 the product of the base and the height (see
figure).
86t 1451
.
t 58
Use a graphing utility to graph T2 with the original
data. How well does the model fit the data?
(d) Evaluate T10 and T20.
(e) Find lim T2. Verify your result using the graph in
t→ part (c).
(f) Interpret the result of part (e) in the context of the
problem. Is it possible to do this type of analysis
using T1? Explain your reasoning.
h
b
(a) Graph the parabolic arch bounded by y 9 x 2
and the x-axis.
(b) Use the limit process to find the area of the
parabolic arch.
(c) Find the base and height of the arch and verify
Archimedes’ formula.
908