52
•
Chapter 1 Prerequisites for Calculus
~
T
•
--
_~.
~~;
~CJ'lapter~:t-~Review Exercises
-::
--
-
r~='
-.
On the Internet, visit
www.phschool.com
for a chapter self-test.
In Exercises 39 and 40, write a piecewise formula for the
function.
In Exercises 1-14, write an equation for the specified line.
1. through (1, -6)
Take it to the NET
with slope 3
39.
2. through (-1, 2) with slope,-1/2
40.
y
y
4. through (-3,6)
(2,5)
5
3. the vertica11ine through (0, -3)
and (1, -2)
-o~--~----~-7X
5. the horizontal line through (0, 2)
6. through (3,3) and (-2, 5)
--~------4----7X
7. with slope -3 and y-intereept 3
8. through (3, 1) and parallel to 2x - y = -2
9. through (4, -12)
and parallel to 4x
+
In Exercises 41 and 42, find
3y = 12
-3) and perpendicular to 3x - 5y
10. through (-2,
11. through (-1,2)
and perpendicular
±x +
to
ty
= 1
where
f has
4
x
-2
2
f(x)
4
2
the following values:
17. y
= x2
16. y =
18. y
- 2x - 1
+
21. y = 1 - cas x
+
1
23. y = -3--2x - x
X4
25. y
=
x
+ cos
x
=
Ix! -
x3
-
-
x
44. f(x)
= \IX,
\ vx
38. y
=
-x - 2
x,
'
(
-x + 2,
-4:5x:50
0<xS4
-2:5x:s:-1
-1<xS1
1<x:52
1
r=t=":
vx + 2
= Vx+l
X2,
and go
f
g(x) = ~
g(x) = ~
a parametrization
is given for a curve.
(b) Find a Cartesian equation for a curve that contains the
parametrized curve. What portion of the graph of the
Cartesian equation is traced by the parametrized curve?
45. x = 5 cas t,
47.
=
x=
48. x
y = 32-x
+
+
2 - t,
= 1
= 2 sin
t,
= 4 sin t,
Y = 11 - 2t,
y
+ t , Y =~,
In Exercises 49-52,
0:5 is:
27T
7T/2:S: t < .37T/2
-2:5
t
i
s: 4
s: 2
give a parametrization
for the curve.
49. the line segment with endpoints (-2, 5) and (4,3)
51. the ray with initial point (2, 5) that passes through (-1, 0)
1
52. y = x (x - 4),
34. y = x2/5
36. y = - 1
4 cas t ,
y
50. the line through (-3, -2) and (4, -1)
-2+~
32. y = tan (2x - 7T)
37.y=l~x-,x,
f)(2)
= ,
g(x)
X,
In Exercises 45-48,
24. y = 1 - sin x
31. y = 2e-x - 3
35. y = In (x -
2 -
~ 2 -
46. x
30.
1
g(x)
43. f(x)
22. Y = see x tan x
28. y=
+ 7T)3) + 1
1
0
(a) Graph the curve. Identify the initial and terminal points,
if any. Indicate the direction in which the curve is traced.
29.y=~
33. y = 2 sin (3x
(d}(g a g)(x)
=
x2
26.y=~
2
f)(x)
42. f(x)
= e-
In Exercises 27-38, find the (a) domain and (b) range, and (c)
graph the function.
27. y
0
= -,
x2/5
20. y = x5
1
(c) (f
41. f(x)
In Exercises 19-26, determine whether the function is even, odd,
or neither.
19. y = x2
(b) (g
In Exercises 43 and 44, (a) write a formula for fog
and find the (b) domain and (c) range of each.
In Exercises 15-18, determine whether the graph of the function
is symmetric about the y-axis, the origin, or neither.
xl/5
g)(- 1)
X
14. through (4, -2) with x-intercept -3
15. y =
0
= 1
12. with x-intercept 3 and y-intercept -5
13. the line y = f(x),
(a) (f
V2 -
x
x:5 2
In Exercises 53 and 54, work in groups of two or three to
(a) fmdf-I and show that (f 0 rl)(x) = (f-I,o f)(x) = x.
(b) graph f and f -I in the same viewing window.
53. f(x)
= 2 - 3x
54. f(x)
= (x
+
2)2,
x::2:-2
Review Exercises
In Exercises 55 and 56, find the measure of the angle in radians
and degrees.
55. sin-I (0.6)
exact answers.
58. Solve the equation sin x ~ -0.2 in the following intervals.
<
(b)
27T
O.2x
59. Solve for x: e-
-00
<x <
60. The graph of f is shown. Draw the graph of each function.
(b) y = -f(x)
(c) y = -2f(x
(d) y = 3f(x - 2) - 2
+ 1) + 1.
(b) How many guppies were present after 4 days? after
I week?
(c) When will there be 2000 guppies?
(d) Writing to Learn Give reasons why this might not be
a good model for the growth of Susan's guppy population.
00
= 4
(a)y=f(-x)
65. Guppy Population The number of guppies in Susan's
aquarium doubles every day. There are four guppies initially.
(a) Write the number of guppies as a function of time t.
.56. tan"" (-2.3)
57. Find the six trigonometric values of () = cas -I (3/7). Give
(a) 0 -s x
53
66. Doctoral Degrees Table 1.22 shows the number of
doctoral degrees earned in the given academic year by
Hispanic students. Let x = 0 represent 1970-71, x = 1
represent 1971-72, and so forth.
y
Table .1,22 Doctorates Earned by Hispanic
Americans
-I
61. A portion of the graph of a function defined on [ - 3, 3] is
shown. Complete the graph assuming that the function is
(a) even.
Source:
(b) odd.
Year
Number of Degrees
1976-77
1980-81
520
460
1984-85
680
1988-89
1990-91
1991-92
1992-93
630
730
810
u.s. Department
830
of Education,
as reported in the Chronicle of
Higher Education, April 28, 1995.
y
(a) Find the linear regression equation for the data and
superimpose its graph on a scatter plot of the data.
(3,2)
(b) Use the regression equation to 'predict the number of
. doctoral degrees that will be earned by Hispanic Americans
in the academic year 2000-01.
.
(c) Writing to learn Find the slope of the regression line.
What does the slope represent?
67. Estimating
62. Depreciation Smith Hauling purchased an 18-wheel truck
for $100,000. The truck depreciates at the constant rate of
$10,000 per year for 10 years.
(3) Write an expression that gives the value y after x years.
(b) When is the value of the truck $55,OOO?
63. Drug Absorption A drug is administered intravenously for
pain. The function
f(
t)
= 90 - 52 In (1 + z},
O::s t :s; 4
gives the number of units of the drug in the body after
t
hours.
Population Growth Use the data in Table 1.23
about the population of New York State. Let x = 60
represent 1960, x = 70 represent 1970, and so forth.
Table 1.23 Population of New York State
Year
Population (millions)
1960
1980
16.78
17.56
1990
17.99
Source: The Statesman's
Press, Ltd., 1992).
Yearbook, 129th ed. (London: The Macmillan
(a) What was the initial number of units of the drug
administered?
(a) Find the exponential regression equation for the data.
(b) How much is present after 2 hours?
(b) Use the regression equation to predict when the
population will be 25 million.
(c) Draw the graph of f
64. Finding Time If Joenita invests $ISOO in a retirement
account that earns 8% compounded annually, how long will
it take this single payment to grow to $5000?
(c) What annual rate of growth can we infer from the
regression equation?
.
62
Chapter 2
Limits and Continuity
Quick Review 2B 1
In Exercises 5-8, write the inequality in the form a
In Exercises 1-4, find [(2).
= 2x3 -
1. [(x)
3. [(x)
= sin
+4
5x2
2. f(x)
=
x
3
-
+
5
4
(7T~)
Ixi < 4
7.lx-21<3
5.
6.
< x < b.
Ixl < c
2
8.lx-cl<d2
In Exercises 9 and 10, write the fraction in reduced form.
X2 - 3x 18
2x2 - x
9.
x + 3
10. -=2--,x2"..--,-+-x----=-1
x<2
3X - 1
_1_'
(
x2 - 1 '
4. [(x) =
4X2
x2:2
Section 2.1 Exercises
In Exercises 1-6, use the graph to estimate the limits and value
of the function, or explain why the limits do not exist.
1.
fT-fH-:
I
I
I
I
I
y
i-
4.
I
I
I,
-f
,
r-
iv!= I x)
I
! I
g-li
:
1:
~+t--,-
!
[(x)
(a) Jim
X-73-
2.
-j-
!
tt
I
F
-
get)
r-TT-' -
]-'-
!
1
!
I
(b) lim
get)
(--7-4+
Y
s4-2
(d) p(-2)
(d) [(3)
I
(a) lim [(h)
I
(c) Jim get)
(---7-4
II
\
I
,
6.
1\
I
J
r
I
,_1::
I
IY
t
---\
I
_\
c. (x
.
.- t-r-
(d) f(O)
t
(a) lim G(x)
(d) G(2)
x
:---
(b) lim G(x)
x---t2-
(c) lim G(x)
--,-j-
f---
A
1
h->O
(d) F(O)
,X---tO
-
r_..L:J.
(c) lim f(h)
(c) Jim F(x)
~<, L
it
I
I
_L_Lk ____
L
h->O+
x---tO+
I
!
L
i
(b) lim [(h)
(b) Jim F(x)
I
III
I
(a) Jim F(x)
x---tO-
-1
I
I
(d) g(-4)
I
I I
if \
t
I
c:r r
h->O-
(c) lim pes)
t
I
i> -,It{ )
I
- - -r--- H- r\
!
1
-t-r{-.
~
i
if
pes)
S--7-2+
I
rr
3.
(b) Jim
I
I I
I
t--7-4-
pes)
. 'Yc-r-.,..-
.J
I
(a) lim
I
~g.t)
1
I ,
f
x---t3
( )
!
I
~l'-.
-1
I
(c) lim[(x)
x---t3+
H- i- -
I
I
s-t-2-
(b) lim [(x)
vi=
c--+- 1/-+-
(a) lim
I:
IS
i
I
I
- II
!
- r-
-~-
I
-1--
!
-r
.-i!
!-
x
1
--±~
r-i-i-tJ
l
I
I
YI
I
r- ---
x---t2+
x-+2
,
In Exercises 7-16, determine the limit by substitution. -Support
graphically.
7. lim
3x2(2x
1)
-
x->-l/2
9. Jim (x3
x->l
+ 3x2
-
2x - 17)
8. lim
x->-4
(x
+ 3)1998
_"
63
Section 2.1 Rates of Change and Limits
y2 + Sy + 6
10. y-->2
lim
+
Y
·
11 .lm
I
2
+ 4y + 3
yZ
y2 - 3
y-->-3
(c) lirn j'(x) = 2
(d) lim f(x)
= 2
(f) lirnf(x)
does not exist.
x41-
X-72
== 1
12. lim int x
13. Jim (x - 6) 2/3
(e) Jim f(x)
14. limYx+3
15. Jim
e' cos x
x....•a
(g) Jim f(x)
== lim I(x)
(h) lim f(x)
exists at every c in (-1, 1).
(i) lim
f(x)
x....•
c
exists at every c in (1,3).
x--....tl+
x~-2
x-->1/2
x ...•2
x-->o·
16. lim In (sin x)
In Exercises 17-20, explain why you cannot use substitution to
determine the limit. Find the limit if it exists.
In Exercises 33-36, match the function with the table.
v:;-=2
17. lim
x...-.+-2
19. lim
_ox
18. Jim ~
x~o x-
M
+ x)2
- 16
x
_0
22. Jim
3(
(2 -
t2 -
I...•
Z
+
,
-
---2+x
5x3 + 8x2
23. x...•
limo 3x 4 - 16x 2
24. Jim
(2 + x)3 - 8
25. lim -'----'---
26. lim sin 2x
.
sin x
27. lim -2-2-x...• 0
x - x
28. lim x
1
2
x
x...•o
x->o
X
+
x ...• 0
29. lim sm x
X
sin x
x
.8
.9
I
1.1
1.2
1.3
In Exercises 31 and 32, which of the statements are true about
the function Y = f(x)
graphed there, and which are false?
=
X
YI
-.4765
-.3111
-.1526
0
.14762
.29091
.43043
~
x+
YI
7.3667
10.8
. 20.9
.8
.9
I
1.1
\.2
1.3
ERROR
-18.9····
-8.8·
-5.367
X
2.7
2.8
2.9
.8
.9
I
1.1
1.2
1.3
ERROR
3.1
3.2
3.~
X= .7
.
,
X= .7
YI
f.IIII!II
30. Jim 3 ~in 4x
x ...• 0 sm 3x
X
31.
36. YI
X= .7
. 2
.r•.••0
+ 1
(a)
x
x2 - X - 2
1
xx2 + x - 2
1
34. YI =
x-I
X
4
1
x2 - 2x
.8
.9
I
1.1
1.2
1.3
2
- 2
x-I
35. YI =
In Exercises 21-30, determine the limit graphically. Confirm
algebraically.
x-I
21. lim -2 --1
x~l
x -
+x
X2
33. Yl =
20. lim (4
x...•o
x ....•o-
x.•.•c
X-->7r12
X-71
(b)
.
YI
-.3
-.2
-.1
ERROR
.1
.2
.3
X~ .7
(c)
(d)
y
y =f(x)
In Exercises 37-44, determine the limit.
37. Jim intx
38. Jim intx
39. Jim intx
40.lim
.
41. lim
42. lim
x...• o-
x-?o+
intx
X---72-
(a)x--t-].f.
lim f(x)
= 1
(b) lim f(x)
x-?o-
(c) lim I(x) = 1
= 0
43. Assume that Jimf(x)
X-70-
X-?O+
(a) lim (g(x)
X""
(e) lirn f(x)
exists
(g) lim f(x)
= 1
x...•
o
(f) Jim f(x)
x...• 0
(h) limf(x)
x...• I
X-70
= 0
(i) Jimf(x)
(j) Jim f(x)
x-?2-
.1HI
32.
-1-1
X
x-?o-
= 0 and Jim g(x)
4
+
3)
(b) lim xf(x)
X-.74
=0
(d) lim
x ...• 4
= 1
44. Assume that Jimf(x)
=
x...•b
2
(a) Jim (f(x)
x ...•b
Y
y = f(x)
-ixi
x
= 3.
x--t4
X---74
(d) lim I(x) = lim I(x)
X-70-
X-7O+
x...•0.0 1
x
g(x)
f(x) - 1
= 7 and lim g(x) = -3.
x->b
+ g(x»
(b) Jim (f(x)
x->b
• g(x»
(d) Jim f(x)
x-eb g(x)
(c) Jim
4 g(x)
x...•b
In Exercises 45-48, complete parts (a), (b), and (c) for the
piecewise-defined function.
(a) Draw the graph of
(b) Determine
2
f
lim, ...•c+ f(x)
and lim; ...•c- f(x).
(c) Writing to Learn Does lim,...•c!(x) exist? If so,
(a) lim
.l:-4-1+
f(x)
= 1
(b) Jimf(x)
x...• z
does not exist.
what is it? If not, explain .
"
-
64
Chapter 2 Limits and Continuity
45. c
= 2,f(x) = ( ~ + I
3 - x
2
46. c
=
2,f(x)
47. c = l,f(x)
=
'
3 - x,
2,
[
x/2,
x<2
Explorations
x>2
In Exercises 59-62, complete the following tables and state
what you believe lim j(x) to be.
x->o
x<2
x=2
x>2
1
= x-I'
x3
-
. x
7Wl
x<l
2x
+
1 - x2,
{ 2,
48.c=-1,f(x)=
(a)
5,
x
7Wl
x=-I
In Exercises 49-52, complete parts (a)-(d) for the piecewisedefined function.
(a) Draw the graph of f
(b) At what points c in the domain of j does limHcf(x)
exist?
(e) At what points c does only the left-hand limit exist?
(d) At what points c does only the right-hand limit exist?
49. j(x) = {sin x,
cas x,
cas X
50. j(x) = {
'
see x,
-Tr :s:: x < 0
0 :s:: x :s:: -tt
. {~'
51. j(x) = 1,
2,
X,
52. j(x) =
[
1,
0,
-0.1
-0.01
-0.001
-0.0001
?
?
?
?
(b)
x ~ 1
x*--l
-2Tr:S:: x < 0
o :s:: x :s:: 2Tr
I
1~~O.~1__ ~O~.O~1 0~.0~0~1 0~.~OO~0~1
__ ~
?
?
?
59. j(x) = x sin l
.
x
l O" - I
61. j(x) = ---x
_
?
l
60. j(x)
= sin
62. j(x)
= x sin (In
e)le
x
Ixl)
e
63. Proof that lime->o (sin
= I when
is Measured in
Radians Work in groups oj two or three. The plan is to
show that the right- and left-hand limits are both 1.
(a) To show that the right-hand limit is 1, explain why we
can restrict our attention to 0<
< Tr/2.
e
(b) Use the figure to show that
area of "'OAP =
Q:s::x< 1
1 :s:: x < 2
x = 2
t
sin B,
area of sector OAP =
area of "'OAT =
e
2'
1
2 tan e.
-I :s:: x < 0, or 0 < x :s:: 1
y
x=O
x < -1, or x> 1
T
In Exercises 53-56, find the limit graphically. Use the Sandwich
Theorem to confirm your answer.
53. lim x sin x
x->o
54. lim x2 sin x
x->o
.
55. lim
x 2' sm2 1
x->o
X
56. lim x cas 2
x->o
X
.
2
1
57. Free Fall A water balloon dropped from a window high
above the ground falls y = 4.9t2 m in t sec. Find the
balloon's
(a) average speed during the first 3 see of fall.
(b) speed at the instant t = 3.
58. Free Fall on a Small Airless Planet A rock released from
rest to fall on a small airless planet falls y = gt? m in t see,
g a constant. Suppose that the rock falls to the bottom of a
crevasse 20 m below and reaches the bottom in 4 sec.
(a) Find the value of g.
(b) Find the average speed for the fall.
(e) With what speed did the rock hit the bottom?
(e) Use (b) and the figure to show that for
1 .
2sm
(d) Show that for 0
written in the form
e <21 e <2I tan. e
< e < 7f/2
e
0<
e < Tr/2,
,
the inequality of (c) can be
1
1 <---<-sin ()
cos ().
Section 2.2
65. Controlling
(e) Show that for 0 < e < 'Tf/2 the inequality of (d) can be
written in the form
sin e
cose<-e-<
(b) Use a graph to estimate an interval (a, b) about
x = 'Tf/6 so that 0.3 < f(x) < 0.7 provided a < x
e
66. Limits and Geometry
Let Pea,
be a point on the
parabola y = X2, a > O. Let 0 be the origin and (0, b) the
y-intercept of the perpendicular bisector of line segment OP
Find Jim p-->o b.
(h) Use (g) to show that
lirn sin
e
e
< b.
a2)
is an even function.
0-70-
< b.
(c) Use a graph to estimate an interval (a, b) about
x = 1i/6 so that 0.49 < f(x) < 0.51 provided a < x
lirn sin e = 1.
(g) Show that (sin e)/e
Let f(x) = sin x.
(a) Find f(rr/6).
1.
(f) Use the Sandwich Theorem to show that
9-->0+
Outputs
65
Limits Involving Infinity
= 1.
(i) Finally, show that
r sin e - 1
9~-e-- .
Extending the Ideas
64. Controlling
Outputs
Let f(x)
=
"\I3x"-=2.
(a) Show that limx-72f(x) = 2 = f(2).
(b) Use a graph to estimate values for a and b so that
1.8 <f(x) < 2.2 provided a < x < b.
(c) Use a graph to estimate values for a and b so that
1.99 <f(x) < 2.01 provided a < x < b.
Limits Involving Infinity
Finite Limits as X~:±:OO • Sandwich Theorem Revisited CD Infinite
Limits as x~a • End Behavior Models III "Seeing" Limits as x~±oo
Finite Limits
asx~±oo
The symbol for infinity (00) does not represent a real number. We use 00 to
describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say "the limit of f as x approaches
infinity" we mean the limit of j as x moves increasingly far to the right on the
number line. When we say "the limit of j as x approaches negative infinity
(- 00)" we mean the limit of j as x moves increasingly far to the left. (The limit
in each case mayor may not exist.)
Looking at j(x) = l/x (Figure 2.9), we observe
(a) as x~oo,
(1/x)~O
=
0,
and we write lim (l/x)
=
and we write lim (l/x)
x"""OO
(b) as x~-oo,
(l/x)~O
O.
X~-CX)
We say that the line y
Definition
[-6,6] by [-4, 4]
Figure 2.9
= 0 is a horizontal asymptote of the graph of f .
--Horiz;;nta'-Asymptot~---------------l
The line y = b is a horizontal asymptote of the graph of a function
= j(x) if either
The graph off (x) = l/x.
y
lim j(x)
L-
z-eee
=b
or
lim j(x)
x.....,-oo
=
b.
\'
I!
.
Section 2.2
Solution
An end behavior model for
=
So, limx--7:,::=f(x)
asymptote of f
71
f is
4x2
2
2x3
X
=0
lirnX-7:'::=2/x
Limits Involving Infinity
and y
=0
is the horizontal
We can see from Examples 8 and 9 that a rational function always has a
simple power function as an end behavior model.
[-10, 10) by [-1, 1)
(a)
"Seeing" Limits as
X--7+oo
We can investigate the graph of y = f(x)
of y = f(l/x)
as x~O.
Example 10
as x~±oo by investigating the graph
USING SUBS~rnUTION
Find lim sin (llx).
X--700
Solution Figure 2.15a suggests that the limit is O.Indeed, replacing
limX--7= sin (I Ix) by the equivalent limx--7o+sin x = 0 (Figure 2.15b),
we find
[-21T, 21T) by [-2,2)
(b)
lim sin l/x
=
X-7C'O
lim sin x
X--7O+
=
O.
Figure 2.15 The graphs of (a) f(x) =
sin (l/x) and (b) g(x) = f(1/x) = sin x.
(Example 10)
Quick Review 2,,2
In Exercises 1-4, find j'"! and graphf, f-1,
same square viewing window.
and y = x in the
= 2x - 3
2. f(x)
=
eX
3. I(x)
=
tan-1 x
4. I(x)
=
coe! x
In Exercises 5 and 6, find the quotient q(x) and remainder rex)
when I(x) is divided by g(x).
5. f(x)
=
-
3:;2
+ x-I,
g(x)
=
3x3
= 2x5 - x3
+ x-I,
g(x) = x3
x2
-
'In Exercises 7-10, write a formula for (a)f(-x)
Simplify where possible.
1. f(x)
2x3
6. f(x)
+ '4x
7. f(x)
= cos x
8. f(x)
9. f(x)
=:
lnx
x
10. f(x)
7. f(x)
=:
+
1
and (b)f(l/x).
=
e-
X
= (x
+ ~ ) sin x
- 5
Section 2.2 Exercises
In Exercises 1-8, use graphs and tables to find (a) limx-->oof(x)
and (b) Jimx-->-oof(x). (e) Identify all horizontal asymptotes.
1. f(x)
= cos (~ )
3. f(x)
=-
5. f(x)
e-X
2. f(x)
= sin 2x
4. f(x)
=
= 3x
lr]
+
1
+2
6. f(x) =
3
3x ,-
[x]
8.f(x)=:~
In Exercises 9-16, use graphs and tables to find the limits.
x
x
x
N
X
x+3
2x - 1
Ix! _ 3
+
1
9. Jim _1_
x...•2+ x - 2
ll.lim
X-7-3-
~3
x
+
10. lim ~2
x...•2- X -
12.lim
X-7-3+
_x_
X
+ ::l
72
Chapter 2 Limits and Continuity
13. lim
intx
x......,O+
37. f(x)
14. lim intx
x->ox
X
15. lim csc x
X-4(ll-l2)+
3
4x
2x
-
+
1
x-2
secx
16.lim
X-7O+
=
38. f(x)
4
+ ~x2
= -x
: x -
3
x In Exercises 17-22, (a) find the vertical asymptotes of the graph
off(x). (b) Describe the behavior off(x) to the left and right of
each vertical asymptote.
I
17. f(x)
= X2 -
4
19. f(x)
= x2 - 2x
=--
20. f(x)
=
22. f(x)
= see x
39. y = eX - 2x
+4
2x
I-x
x+I
21. f(x)
behavior model for the function.
1
X2 -
18·f(x)
= cotx
In Exercises 39-42, find (a) a simple basic function as a right
end behavior model and (b) a simple basic function as a left end
41. y = x
2x2 - Sx - 3
+ In Ixl
40. y = x2
+ e-
42. y = x2
+
X
sin
In Exercises 43-46, use the graph of y = f(lIx)
limx->oof(x) and limx->-oof(x).
x
to find
In Exercises 23-28, find 'Jimx-->=y and Jimx->-= y.
43. f(x)
= xe?
44. f(x)
= x2 e-x
23. y= (2 __ x )(~)
x + I S
45. f(x)
= In Ixl
46. f(x)
= x sin
25
+ x2
= cos (l/x)
27
•y
=
In Exercises 47 and 48, find the limit of f(x) as (a)
(b) X-700, (c) X-70-, and (d) X-70+,
sin x
x
28
sin x
2x2
+
26. y = 2x
I + (l/x)
• y
x
+X
=
+ 2 sin x
x sin x
. y
47. f(x) =
2x2
In Exercises 29-32, match the function with the graph of its end
behavior model.
2x3 - 3x2
29.y='=':':"'-~~':"
X'+3
31. y =
2X4
-
x3
+
I
30. y
=
x
+ +1
~--=-?
+x - 3
S -
x4
X
=---~~.:..
2x-
+ X2
-
1
---'-'-~~---=2-x
X4
+ x2
3x3
-
32. y =
~
2
I-x
1
-I,
X-7-00,
x2:0
I
x- 2
-48. f(x) = x-I'
I/x2,
x<O
x> 0
In Exercises 49 and SO,work in groups of two or three and
sketch a graph of a function' y = f(x) that satisfies the stated
conditions. Include any asymptotes,
x~l
lim f(x)
x---too
lim
x---t-2-
f(x)
50. Jim f(x)
x---t2
(b)
x
x <0
(lIx,
49. lim f(x)
(a)
.l
lim f(x)
x---too
=
2,
= -I,
=
Jim f(x) =
lim f(x)
00,
x~5-
00,
x~5+
Iim f(x)
='
lim f(x)
=0
X--7-2+
=
00,
-00,
X-7-00
= -1,
=
00,
lim f(x)
X---74+
lim f(x)
X--7-00
=
-00,
lim f(x)
x---t4-
=
00,
= 2
51. End Behavior Models Work in groups of two or three,
Suppose that gl (x) is a right end behavior model for I,(x)
and that g2(X) is a right end behavior model forf2(x),
Explain why this makes gl (X)/g2(X)
a right end behavior
model for fl(X)/h(x).
(el
(d)
In Exercises 33-38, (a) find a power function end behavior
model for f (b) Identify any horizontal asymptotes.
33. f(x)
= 3x2 - 2x
34. f(x)
= -4x3
35. f(x)
x-2
= -2---'x2::-+-3-x---S-
+
+ x2
1
- 2x - 1
36. f(x)
2
= 3x
x2
X
-
+
4
S
52. Writing to Learn Let L be a real number, limx->cf(x) = i,
and limx->c g(x) = 00 or -00. Can limx->c (J(x) + g(x» be
determined? Explain,
Section 2.3
Exploration
53. Table 2.2 gives the average amount per grant of emergency financial assistance for veterans in Franklin County,
Ohio. Let .r =
represent 1980, .r = 1 represent 1981,
and so forth.
°
54. Exploring Properties of Limits Find the limits of f, g, and
fg as X--7C.
1
(a)f(x) = -,
g(x) = x,
x
Table 2.2 Emergency Financial Assistance
Year
73
Continuity
2
(b)f(x)
. Average Amount
(dollars)
(c)f(x)=
= -3'
x
3
C
=
g(x) = 4x3,
x-2'
°
C
= 0
g(x)=(x-2)3,
c=2
1986
145.38
1987
1988
155.00
230.43
(d)f(x)
1989
1990
1991
1992
1993
1994
420.70
494.55
555.00
508.77
460.92
453.77
(e) Writing to Learn Suppose that limx->c!(x) = and
lim,He g(x) = 00. Based on your observations in (a)-(d),
what can you say about limx->e (J(x) • g(x»?
III
Source: Franklin County Veterans Service Commission
= (3 ~
X)4'
g(x) = (x - 3)2,
C
= 3
°
Extending the Ideas
55. The Greatest Integer Function
as reported by
(a) Show that
Mary Stephens in The Columbus [Ohio} Dispatch, May 31, 1995.
1
int x
1 (x>O)
x
.
. intx
(b) Determine lim --.
x-+oo
x
X
---<--:5
(a) Find a cubic regression equation and superimpose its
graph on a scatter plot of the data.
x
(b) Find a quartic regression equation and superimpose its
graph on a scatter plot of the data.
(c) Use each regression equation to predict the average
amount per grant of financial assistance for veterans in the
year 2000.
(d) Writing to Learn Find an end behavior model for
each regression equation. What does each predict about the
future sizes of grants? Does this seem reasonable? Explain.
x-I
int x
)
and -->--;::=:
1 (x<O.
x
x
.
.
intx
(c) Determine lim --.
x-+-oo
X
56. Sandwich Theorem Use the Sandwich Theorem to
confirm the limit as X--7OO found in Exercise 3.
•
57. Writing to Learn Explain 'why there is no value L for
which limx-->~ sin x = L.
In Exercises 58-60, find the limit. Give a convincing argument
that the value is correct.
58. lim In x2
x-->~
lnx
59. lim
x-->""
In x
log x
60. lim In (x + 1)
x-->~
lnx
200
190
180
~c
E
?l
'""
Continuity
170
Continuity at a Point e Continuous Functions • Algebraic
Combinations " Composites 0 Intermediate Value Theorem for
Continuous Functions
160
150
eo
140
130
'fd 120
::c:" 110
100
90
80
e
Continuity at a Point
Figure 2.16 How the heartbeat returns to
a normal rate after running.
When we plot function values generated in the laboratory or collected in the
field, we often connect the plotted points with an unbroken curve to show
what the function's values are likely to have been at the times we did not measure (Figure 2.16). In doing so, we are assuming that we are working with
a continuous function, a function whose outputs vary continuously with the
inputs and do not jump from one value to another without taking on the
values in between. Any function y = f(x) whose graph can be sketched in one.
continuous motion without lifting the pencil is an example of a continuous
function.
o
Minutes after exercise
80
Chapter 2 Limits and Continuity
Quick Review 2.3
3x2
1. Find x....•
lim-l
2. Let f(x)
=
(a) lim
3
+4
(b) Jim f(x)
x....•
-I+
.
+
5,
{X2 -
3. Let f( x ) -
5. f(x)
.
4x
4 _ x,
=
x2,
(g
0
(c) Jim f(x)
(d) f(-I)
x--.-I
(g
+
+
g =' [0,00)
x> 0
9x - 5 = O.
2x - 1 = O.
In Exercises 9 and 10, let
(c) Urnf(x)
x--t2+
domain of
= l/x,
f)(x)
x3
8. Use graphing to solve
x <2
x_> 2 .
(b) lim f(x)
x~2-
0
sin x2,
7. Use factoring to solve 2x2
Find each limit.
(a) lim f(x)
=
f) (x)
6. g(x) =~,
int x. Find each limit.
f(x)
x....•
-I-
+1
2x
X
f(x)
(d)f(2)
x-+2
= (
~:z
x~ 6x - 8,
~ ~ ~.
9. -Solve the equation f(x) = 4.
In Exercises 4-6, find the remaining functions in the list of
functions: t. g, fog,
g 0f
4. f(x)
2x - 1
x+5
= --,
g(x)
1
x
=- +
10. Find a value of c for which the equation f(x)
= c has no
solution.
1
Section 2.3 Exercises
In Exercises 1-10, find the points of discontinuity of the
function. Identify each type of discontinuity.
2)2'
x + 1
2. y = x2 - 4x + 3
1
4. y =
1
1. y = (x
+
1
3. y = x2
+
6. y = V2x - 1
5.y=~
7. y
=
Ix - 11
8. y = cotx
Ixl/x
10. y = In (x + 1)
9. y = e1/x
In Exercises 11-18, use the functionfdefined
below to answer the questions.
f(x)
x2 - 1,
2x,
1,
-2x + 4,
0,
=
1
and graphed
-l:O;x<O
O<x<l
x=1
l<x<2
2<x<3
y
12. (a) Does f(l)
exist?
(b) Does lim,....•
1 f(x)
exist?
(c) Does lirnx->d(x)
= f(1)?
f continuous
Is f defined at
(d) Is
13. (a)
at x =·1?
x = 2? (Look at the definition of f)
(b) Is f continuous at x = 2?
14. At what values of x is f continuous?
15. What value should be assigned to f(2)
function continuous at x = 2?
to make the extended
16. What new value should be assigned to f(l) to make the new
function continuous at x = I?
17. Writing to Learn Is it possible to extendfto be
continuous at x = O? If so, what value should the extended
function have there? If not, why not?
18. Writing to Learn Is it possible to extend f to be
continuous at x = 3? If so, what value should the extended
function have there? If not, why not?
Y=f(X;~
In Exercises 19-24, (a) find each point of discontinuity.
(b) Which of the discontinuities are removable? not removable?
Give reasons for your answers.
y=2x
1
--~--~--~----~--~--~x
19. f(x)
= (~ ~ Xl
2
-1
20. f(x) = {;,- x
11. (a) Does f(-l)
exist?
(b) Does limx->-l+ f(x)
x/2,
: : ~
'
x<2
x=2
x>2
exist?
(c) Does lim,....•
-I+ f(x)
= f(-I)?
(d) Is f continuous at x = -I?
21. f(x)
= ( x ~ 1'
x3 - 2x + 5,
x<1
X 2:: 1
Section 2.3
I22. f(x)
x =1--1
X2
= {2,
'
41. Continuous Function
function
x =-1
Find a value for a so that the
I(x) =
y
23.
81
Continuity
{X2 -
1,
2ax,
y;f(x)
x<3
X 2:::
3
is continuous.
42. Writing to learn
Explain why the equation
e-
X
=
x has
at least one solution.
43. Salary Negotiation A welder's contract promises a 3.5%
salary increase each year for 4 years and Luisa has an initial
salary of $36,500.
y
24.
y =f(x)
(a) Show that Luisa's salary is given by
y = 36,500(1.035)intt,
where t is the time, measured in years, since Luisa signed
the contract.
__J-__~
~x
~ __-L __ ~
(b) Graph Luisa's salary function. At what values of t is it
continuous?
-1
In Exercises 25-30, give a formula for the extended function that
is continuous at the indicated point.
X2
25. f(x)
27. f(x)
-
9
= --.;-:t3'
sin x
= -,
x
x
x
= -3
=0
x-4
29. f(x)
= , I
30. f(x)
=
''
vx - 2
x3
-
26. f(x)
X =
4x2 - llx
x2 _ 4
28. f(x)
=
x3 - 1
x2 _ I '
sin 4x
= --,
=1
x
x
x
=0
4
+
30
,x
44. Airport Parking Valuepark charge $1.10 per hour or
fraction of an hour for airport parking. The maximum charge
per day is $7.25.
(a) Write a formula that gives the charge for
o ::; x
(b) Graph the function in (a). At what values of x is it
continuous?
Exploration
=2
45. Let f(x)
In Exercises 31-34, work in groups of two or three. Verify that
the function is continuous and state its domain. Indicate which
theorems you are using, and which functions you are assuming
to be continuous.
31. Y = , r=r":
vx + 2
32. y = x2
33. y = Ix2 - 4xl
34. y =
+ V4 -
x
x2 - 1
{
--.;=T'
x
2,
x = I
In Exercises 35-38, sketch a possible graph for a function
has the stated properties.
35. f(3)
exists but limx->3f(x)
36. f(-2)
exists, limx->-2+ f(x)
does not exist.
37. f(4) exists, limx->4f(x)
x = 4.
=I-
1
I that
does not.
= f(-2),
exists, butfis
x hours with
::; 24. (Hint: See Exercise 43.)
but limx---+_2f(x)
= (1 + ~
r-
(a) Find the domain of
(b) Draw the graph of
f
f
(e) Writing to learn Explain why x = -1 and x = 0
are points of discontinuity of f
(d) Writing to learn Are either of the discontinuities
in (c) removable? Explain.
(e) Use graphs and tables to estimate
Extending the Ideas
46. Continuity at a Point
x = a if and only if
Show that j'(x) is continuous at
lim f(a
h..•o
not continuous at
38. f(x) is continuous for all x except x = 1, where f has a
nonremovable discontinuity.
39. Solving Equations Is any real number exactly 1 less than
its fourth power? Give any such values accurate to 3 decimal
places.
40. Solving Equations Is any real number exactly 2 more than
its cube? Give any such values accurate to 3 decimal places.
lirnx->= f(x).
+ h)
= f(a).
47. Continuity on Closed Intervals Letfbe
never zero on [a, b]. Show that either f(x)
[a, b] or f(x) < 0 for all x in [a, b].
48. Properties of Continuity
an interval, then so is If I.
continuous and
> 0 for all x in
Prove that if f is continuous on
49. Everywhere Discontinuous
Give a convincing argument
that the following function is not continuous at any real
number.
f(x)
=
{I,
0,
if x ~s ~atio.nal
if x 1S irrational
Ii;l!
Section 2.4
Rates of Change and Tangent Lines
= f(t). Its instantaneous speed at any time t is the
instantaneous rate of change of position with respect to time at time t, or
of change of its position y
Particle Motion
We only have considered
objects
moving in
one direction in this chapter. In Chapter 3,
l'
we will deal with more complicated
h~
motion.
+ h) - f(t)
f(t
h
.
We saw in Example 1, Section 2.1, that the rock's instantaneous
t = 2 see was 64 ft/sec.
Example 6
= 16t2.
The position function of the rock is f(t)
speed of the rock over the interval between t = 1 and t
Solution
f(1
+ h)
- f(1)
16(1
+ h)2
h
=
=
32 ft/sec.
6. through (1, 6) and (4, - I)
A to point B.
2. A(l, 3),
B(a, b)
In Exercises 3 and 4, find the slope of the line determined by the
points.
3. (-2,3),
(5, -I)
4. (-3, -1),
(3,3)
In Exercises 5-9, write an equation for the specified line.
7. through (I, 4) and parallel to y =
3
-"4x
+2
8. through (1, 4) and perpendicular to' y = - ~x
9. through (-1, 3) and parallel to 2x + 3y = 5
Section 2.4 Exercises
1. f(x)
= x3
+
I
2·f(x)=~
(a) [0, 2}
3. f(x)
(b)
uo,
a}
(a) [I, 4}
(b) [I, 3}
(b)
nco,
103}
= cot t
(a) [7T/4, h/4}
6. f(x)
12}
= e'
(a) [ -2,
4·f(x)=lnx
S. f(x)
(b)[-I, I)
(a)[2,3)
+2
10. For what value of b will the slope of the line through (2, 3)
and (4, b) be 5/3?
S. through (-2, 3) with slope = 3/2
In Exercises 1-6, find the average rate of change of the function
over each interval.
The average
1 + h sec was
1 was
+ 2)
lim 16(h
B(3,5)
1 sec.
- 16(1)2
h-..;O
In Exercises 1 and 2, find the increments ~x and ~y from point'
=
=
h
The rock's speed at the instant t
2),
speed at
INVESTIGATING FREE FALL
Find the speed of the falling rock in Example 1, Section 2.1, at t
1. A(-5,
87
=2
(a)[O,7T)
+
(b) [7T/6, 7T/2}
cas t
(b)[-7T,7T)
88
Chapter 2 Limits and Continuity
In Exercises 9-12, at the indicated point find
In Exercises 7 and 8, a distance-time graph is shown.
(a) the slope of the curve,
(a) Estimate the slopes of the secants PQI. PQ2. PQ3. and
PQ4. arranging them in order in a table. What is the
appropriate unit for these slopes?
(b) an equation of the tangent, and
(c) an equation of the normal.
P
(b) Estimate the speed at point
(d) Then draw a graph of the curve, tangent line, and normal
line in the same square viewing window.
7. Accelerating from a Standstill The figure shows the
distance-time graph for a 1994 Ford® Mustang Cobra"
accelerating from a standstill.
9. y = x2
x = -2
at
10. y = x2 - 4x
s
650
600
11. Y
12. y
x = 1
at
1
= --1
x= x2 - 3x
at
x
- 1
=
2
at
x
=
0
500
In Exercises 13 and 14,find the slope of the curve at the
indicated point.
:§: 400
<l
u
c:
13. f(x)
~ 300
is
=
[x]
(a) x = 2
at
14.f(x)=ix-2i
200 f---+--.,<i
at
(b)x
=
-3
x=l
In Exercises 15-18, determine whether the curve has a tangent
at the indicated point. If it does, give its slope. If not, explain
why not.
100
5
10
15
20
Elapsed time (see)
8. Lunar Data The figure shows a distance-time graph for a
wrench that fell from the top platform of a communication
mast on the moon to the station roof 80 m below.
•
2 - 2x - x2
+ 2.
'x
15. f(x)
= { 2x
16. f(x)
= { 2'
x - x,
17. f(x)
=
-x
18. f(x)
4 - x
4
= {Sin x,
cas x,
x<O
x2:0
<0
2:
0
at
x =0
at
x = 0
at
x
x:o;2
I/x,
I
X
x> 2
= 2
'
O:o;x<3wM
3w/4 :0; x :0; 2w
~
x=3wM
In Exercises 19-22, (a) find the slope of the curve at x = a and
(b)·Writing to Learn describe what happens to the tangent at
x = a as a changes.
19. y
21 y
.
80
60 f-- 1--
::§
40
<l
o
c:
S
'"
is
20
0
c,ljP
I
c:
~
1
= --x-I
20. Y
= 2/x
22. Y
=9-
X2
23. Free Fall An object is dropped from the top of a lOO-m
tower. Its height above ground after t sec is 100 - 4.9t2 m,
How fast is it falling 2 see after it is dropped?
y
:§:
= X2 + 2
V
I
~
rr:I
Tffz
f--
! I
i
/.
./
I IQd/ .
!y
L.-- 1---1 I
r5
Elapsed time (see)
!~
~ \--.
"-
\--
f- j--
10
24. Rocket Launch At t see after lift-off, the height of a rocket
is 3t2 ft. How fast is the rocket climbing after 10 see?
25. Area of Circle What is the rate of change of the area of a
circle with respect to the radius when the radius is r = 3 in.?
26. Volume of Sphere What is the rate of change of the
volume of a sphere with respect to the radius when the
radius is r = 2 in.?
Section 2.4
27. Free Fall on Mars The equation for free fall at the surface
of Mars is s = 1.86t2 m with t in seconds. Assume a rock
is dropped from the top of a 200-m cliff. Find the speed of
the rock at t = 1 sec.
Rates of Change and Tangent Lines
89
Table 2.3 Federal Funding
Year
INS Funding
($ billions)
1993
1.5
1994
1.6
1995
2.1
1996
2.6
1997
3.1
Source: Immigration
and Naturalization
Service as reported by Bob Laird
in USA Today, February 18, 1997.
34. Table 2.4 gives the amounts of money earmarked by the U.S.
Congress for collegiate academic programs for several years.
(a) Let x = 0 represent 1980, x = 1 represent 1981, and
so forth. Make a scatter plot of the data.
(b) Let P represent the point corresponding to 1997, and Q
the point for anyone of the previous years. Make a table of
the slopes possible for the secant line PQ.
28. Free Fall on Jupiter The equation
surface of Jupiter is s = 11.44t2 m
Assume a rock is dropped from the
Find the speed of the rock at t = 2
for free fall at the
with t in seconds.
top of a 500-m cliff.
sec.
29. Horizontal Tangent At what point is the tangent to
f(x) = x2 + 4x - 1 horizontal?
(c) Writing to Learn Based on the computations, explain
why someone might be hesitant to make a prediction about
the rate of change of congressional funding in 1997.
Table 2.4 Congressional Academic Funding
Funding
30. Horizontal Tangent At what point is the tangent to
f(x) = 3 - 4x - X2 horizontal?
Year
($ millions)
31. Finding Tangents and Normals
1988
1989
225
289
1990
1991
1992
1993
1994
1995
1996
·270
493
684
763
651
(a) Find an equation for each tangent to the curve
y = l/(x - 1) that has slope -1. (See Exercise 21.)
(b) Find an equation for each normal to the curve
y = l/(x - 1) that has slope 1.
32. Finding Tangents Find the equations of alllines tangent
to y = 9 - X2 that pass through the point (1, 12).
33. Table 2.3 gives the amount of federal funding for the
Immigration and Naturalization Service (INS) for
several years.
600
296
1997
446
Source: The Chronicle of Higher Education,
March 28, 1997.
(a) Find the average rate of change in funding from 1993
to 1995.
Explorations
(b) Find the average rate of change from 1995 to 1997.
In Exercises 35 and 36, complete the following for the function.
(c) Let x = 0 represent 1990, x = 1 represent 1991, and
so forth. Find the quadratic regression equation for the data
and superimpose its graph on a scatter plot of the data.
(a) Compute the difference quotient
(d) Compute the average rates of change in (a) and (b) using
the regression equation.
(b) Use graphs and tables to estimate the limit of the
difference quotient in (a) as h~O.
(e) Use the regression equation to find how fast the funding
was growing in 1997.
(c) Compare your estimate in (b) with the given number.
f(1
+
h) - f(1)
h
(d) Writing to Learn Based on your computations,
do you think the graph of f has a tangent at x = I? If so,
estimate its slope. If not, explain why not.
36. f(x)
= 2X,' In 4
II
90
Chapter 2
Limits and Continuity
Extending the Ideas
In Exercises 37-40, work in groups of two or three.
The curve y
= f(x)
has a vertical tangent at x
I
hl!Po
f(a
+ h)
- f(a)
=
=a
if
OQ
h
2
or if
lim f(a
h-->O
In Exercises 41 and 42, determine whether the graph of the
function has a tangent at the origin. Explain your answer.
+
h) - f(a)
41. f(x)
=
[
-00.
h
In each case, the right- and left-hand limits are required to be the
same: both +00 or both -00.
42. f(x)
=
= x2/5
38. y
= x3/5
39. y = xll3
40. y
= X2/3
37. y
1
[x sin
x=t-o
x=o
0,
0,
Use graphs to investigate whether the curve has a vertical tangent
at x = O.
.
x sm~,
=
±,
x =t- 0
x = 0
43. Sine Function Estimate the slope of the curve y = sin x
'at x = 1. (Hint: See Exercises 35 and 36.)
Key Terms
average rate of change (p. 82)
average speed (p. 55)
connected graph (p. 79)
Constant Multiple Rule for Limits (p. 67)
continuity at a point (p. 75)
continuous at an endpoint (p. 75)
continuous at an interior point (p. 75)
continuous extension (p. 77)
continuous function (p. 77)
continuous on an interval (p. 77)
difference quotient (p. 86)
Difference Rule for Limits (p. 67)
discontinuous (p. 75)
end behavior model (p. 70)
free fall (p. 55)
horizontal asymptote (p. 65)
infinite discontinuity (p. 76)
instantaneous rate of change (p. 87)
instantaneous speed (p. 87)
intermediate value property (p. 79)
Intermediate Value Theorem for
Continuous Functions (p, 79)
jump discontinuity (p. 76)
left end behavior model (p. 69)
left-hand limit (p. 60)
limit of a function (p. 57)
normal to a curve (p. 86)
oscillating discontinuity (p. 76j
point of discontinuity (p. 75)
Power Rule for Limits (p. 67)
Product Rule for Limits (p. 67)
Properties of Continuous Functions (p, 78)
Quotient Rule for Limits (p. 67)
removable discontinuity (p. 76)
right end behavior model (p. 69)
right-hand limit (p. 60)
Sandwich Theorem (p, 61)
secant to a curve (p, 82)
slope of a curve (p. 85)
Sum Rule for Limits (p. 67)
tangent line to a curve (p. 85)
two-sided limit (p. 60)
vertical asymptote (p. 68)
vertical tangent (p. 90)
Review Exercises
91
Take it to the NET
On the Internet. visit
www.phschool.com
Review Exercises
In Exercises 1-14, find the limits.
1. lim
x...•;-2
(X3 -
+
2X2
2.1im
1)
x->-2
V1="2x"
4
1
5.lim
+x
2
x->o
12x3
+ x3
+
128
x csc x + 1
9.lim---x...•o
x csc x
11. lim
1
+5
25. Determine
(a) lim g(x).
X2
(b) g(3).
x~3-
(c) whether g(x) is continuous at x = 3.
6 li
X
X4
7'x~~oo
2x
In Exercises 25 and 26, use the graph of the function with
domain - 1 ::s x ::s 3.
x-t5
1
2
----
+
-
4. lim "V"9 -
3. lim
.r•.•
x2
3x2
for a chapter self-test.
2X2
+3
+7
'x-!~oo
5x2
8.lim- .<->04
sin 2x
x
(d) the points of discontinuity of g(x).
(e) Writing to Learn whether any points of discontinuity
are removable. If so, describe the extended function. If not,
explain why not.
10. lim eX sin x
y
x->o
int (2x - I)
12. lim
x->712-
x->7/2+
int (2x - 1)
14. lim x
x...•oo x
13. lim e- cas x
X
x-too
2
+ sin x
+ cas x
In Exercises IS-20, determine whether the limit exists on the
basis of the graph of y = f(x). The domain of f is the set of real
numbers.
15. lim
f(x)
x...•d
16. Iim f(x)
17. lim f(x)
18. timf(x)
--'---,-+---'----'----'--+x
26. Determine
X-4C+
x-tC-
(a) lim k(x).
(b) lim k(x).
X-41-
(c) k(l).
X---7I+
.t-4C
19. limf(x)
(d) whether k(x) is continuous at x = 1.
20. limf(x)
x...•b
x->a
(e) the points of discontinuity of k(x).
y
e
(f) Writing to Learn whether any points of discontinuity
are removable. If so, describe the extended function. If not,
explain why not.
y=fcx
y
y
---r-L---~-L--~--~x
a
In Exercises 21-24, determine whether the functionfused
Exercises 15-20 is continuous at the indicated point.
21. x
=a
23. x = c
22. x
2
= k(x)
(1. l.5)
in
=b
24. x = d
In Exercises 27 and 28, (a) find the vertical asymptotes of the
graph of y = f(x), and (b) describe the behavior of f(x) to the
left and right of any vertical asymptote.
27. f(x)
x+3
= x
+2
x-I
28. f(x)
= x2 (x
+ 2)
92
Chapter 2 Limits and Continuity.
In Exercises 29 and 30, answer the questions for the piecewisedefined function.
1.
x:O;-l
~ x~
1,
x~l
39. f(x)
(
(b) Does f have a limit as x approaches -I?
what is it? If not, why not?
(c) Is f continuous at x = -I?
= {lx3 X2
4xl,
2x - 2,
-
x
x
O? I? If so,
=
(
f
(b) Does
X--73+
1
have a limit as x--7l ? If so, what is it? If not,
In Exercises 31 and 32, find all points of discontinuity of the
function.
+
32. g(x) = V3x
31 • f( x )=~ 4 _ x2
2
In Exercises 33-36, find (a) a power function end behavior
model and (b) any horizontal asymptotes.
2x + 1
x2 - 2x + 1
3
= x
-
4x2 + 3x
x-3
34. f(x)
= 2x2 + 5x - 1
x2
+3
36. f(x)
+
2x
2
= x4 - 3x
x3
-
X
+ x-I
+1
In Exercises 37 and 38, find (a) a right end behavior model and'
(b) a left end behavior model for the function.
37. f(x)
= x + eX
38. f(x)
xr 0
k,
x = 0
= 3,
42. lim f(x)
X--72
lim f(x)
=
00,
X--7-00
=
00,
lim f(x)
X--73-
does not exist,
=
-00
lim f(x)
X---72+
= 1(2) = 3
43. Average Rate of Change Find the average rate of change
of I(x) = 1 + sin x over the interval [0, 1T/2].
44. Rate of Change Find the instantaneous rate of change of
the volume V = (l/3)1Tr2H
of a cone with respect to the
radius r at r = a if the height H does not change.
I continuous?
is I discontinuous?
(c) At what points is
(d) At what points
lim f(x)
lim f(x)
<1
2:
lx'
x~oo
why not?
35. f(x)
'*
3
x =3
In Exercises 41 and 42, work in groups of two or three. Sketch
a graph of a function f that satisfies the given conditions.
41.
O? I? Explain.
(a) Find the right-hand and left-hand limits of f at x = 1.
=
k,
SIll X
40. f(x)
(a) Find the right-hand and left-hand limits of j at x = -1,
0, and 1.
33. f(x)
- 15
x - 3
,x
=
O<x<l
\
30. f(x)
+ 2x
X2
-l<x<O
x=O
-x
29. f(x) =
In Exercises 39 and 40, work in groups of two or three. What
value should be assigned to k to make f a continuous function?
=
In
Ixl + sin x
45. Rate of Change Find the instantaneous rate of change of
the surface area S = 6x2 of a cube with respect to the edge
length x at x = a.
46. Slope of a Curve Find the slope of the curve
y = x2 - X - 2 at x = a.
47. Tangent and Normal Let I(x) = X2 - 3x and
P = (1, f(1».
Find (a) the slope of the curve y = f(x)
at P, (b) an equation of the tangent at P, and (c) an equation
of the normal at P
48. Horizontal Tangents At what points, if any, are the
tangents to the graph of I(x) = x2 - 3x horizontal?
(See Exercise 47)
49. Bear Population The number of bears in a federal wildlife
reserve is given by the population .equation
200
pet)
= 1
+
7e-01t'
where t is in years.
(a) Writing to Learn Find p (0). Give a possible
interpretation of this number.
(b) Find lim pet).
1-'0=
(c) Writing to Learn Give a possible interpretation
result in (b).
of the
Review Exercises
50. Taxi Fares Bluetop Cab charges $3.20 for the first mile and
$1.35 for each additional mile or pan of a mile.
(a) Write a formula that gives the charge for x miles with
o :s; x :s; 20.
51. Congressional Academic Funding
Exercise 34 of Section 2.4.
Consider Table 2.4 in
(a) Let x = 0 represent 1980, x = 1 represent 1981, and
so forth. Find a cubic and a quartic regression equation for
the data.
(b) Find an end behavior model for each regression
equation. What does each predict' about future funding?
52. Limit Properties
53. Table 2.5 gives the spending for software, equipment, and
services to create and run intranets-private, Internet-like
networks that link a company's operations.
a
x = represent 1990, x = 1 represent 1991, and
so forth. Make a scatter plot of the data.
(a) Let
(b) Graph the function in (a). At what values of x is it
discontinuous?
(b) Let P represent the point corresponding to 2000, and Q
the point for anyone of the previous years. Make a table of
the slopes possible for the secant line PQ.
(c) Predict the rate of change of spending in 2000.
(d) Find a quadratic regression equation for the data, and
use it to calculate the rate of change of spending in 2000.
Table 2.5 Intranet Spending
Assume that
Spending
lim [J(x)
x->c
+ g(x)]
= 2,
lim [J(x)
x->c
- g(x)]
= 1,
and that limx->cf(x)
and limx->cg(x)
limx->cf(x)
and limx->c g(x).
Year
exist. Find
93
1995
1996
1997
1998
1999
2000
($ billions)
2.7
4.8
7.8
11.2
15.2
20.1
Source: Killen & Associates as reported by Anne R. Carey and Elys A.
McLean in USA Today. April 14, 1997.