Section 2.1 The Idea of Limits
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1
Limits
Chapter Preview All of calculus is based on the idea of a limit. Not only are limits important in their own
right, but they underlie the two fundamental operations of calculus: differentiation (calculating derivatives) and integration
(evaluating integrals). Derivatives enable us to talk about the instantaneous rate of change of a function, which, in turn, leads
to concepts such as velocity and acceleration, population growth rates, marginal cost, and flow rates. Integrals enable us to
compute areas under curves, surface areas, and volumes. Because of the incredible reach of this single idea, it is essential to
develop a solid understanding of limits. We first present limits intuitively by showing how they arise in computing instantaneous velocities and finding slopes of tangent lines. As the chapter progresses, we build more rigor into the definition of the
limit, and we examine different ways in which limits arise. The chapter concludes by introducing the important property
called continuity and by giving the formal definition of a limit.
2.1 The Idea of Limits
This brief opening section illustrates how limits arise in two seemingly unrelated problems: finding the instantaneous velocity
of a moving object and finding the slope of a line tangent to a curve. These two problems provide important insights into
limits on an intuitive level. In the remainder of the chapter, we develop limits carefully and fill in the mathematical details.
Average Velocity
Instantaneous Velocity
Slope of the Tangent Line
Quick Quiz
SECTION 2.1 EXERCISES
Review Questions
1.
Suppose sHtL is the position of an object moving along a line at time t ¥ 0. What is the average velocity between
the times t = a and t = b?
2.
Suppose sHtL is the position of an object moving along a line at time t ¥ 0. Describe a process for finding the
instantaneous velocity at t = a.
3.
What is the slope of the secant line between the points Ha, f HaLL and Hb, f HbLL on the graph of f ?
4.
Describe a process for finding the slope of the line tangent to the graph of f at Ha, f HaLL.
5.
Describe the parallels between finding the instantaneous velocity of an object at a point in time and finding the
slope of the line tangent to the graph of a function at a point.
6.
Graph the parabola f HxL = x2 . Explain why the secant lines between the points H-a, f H-aLL and Ha, f HaLL have
slope zero. What is the slope of the tangent line at x = 0?
Basic Skills
7.
Average velocity The function sHtL represents the position of an object at time t moving along a line. Suppose
sH2L = 136 and sH3L = 156. Find the average velocity of the object over the interval of time @2, 3D.
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Chapter 2 • Limits
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9.
Average velocity The function sHtL represents the position of an object at time t moving along a line. Suppose
sH1L = 84 and sH4L = 144. Find the average velocity of the object over the interval of time @1, 4D.
Average velocity The position of an object moving along a line is given by the function sHtL = -16 t2 + 128 t.
Find the average velocity of the object over the following intervals.
a. @1, 4D
b. @1, 3D
c. @1, 2D
d. @1, 1 + hD, where h > 0 is a real number
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10. Average velocity The position of an object moving along a line is given by the function
sHtL = -4.9 t2 + 30 t + 20. Find the average velocity of the object over the following intervals.
a. @0, 3D
b. @0, 2D
c. @0, 1D
d. @0, hD, where h > 0 is a real number
11. Average velocity The table gives the position sHtL of an object moving along a line at time t, over a two-second
interval. Find the average velocity of the object over the following intervals.
a. @0, 2D
b. @0, 1.5D
c. @0, 1D
d. @0, 0.5D
t
0
0.5
1
1.5
2
sHtL
0
30
52
66
72
12. Average velocity The graph gives the position sHtL of an object moving along a line at time t, over a 2.5-second
interval. Find the average velocity of the object over the following intervals.
a. @0.5, 2.5D
b. @0.5, 2D
c. @0.5, 1.5D
d. @0.5, 1D
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13. Average velocity Consider the position function sHtL = -16 t2 + 100 t representing the position of an object
moving along a line. Sketch a graph of s with the secant line passing through H0.5, sH0.5LL and H2, sH2LL.
Determine the slope of the secant line and explain its relationship to the moving object.
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Section 2.1 The Idea of Limits
14. Average velocity Consider the position function sHtL = sin p t representing the position of an object moving
along a line on the end of a spring. Sketch a graph of s with the secant line passing through
H0, sH0LL and H0.5, sH0.5LL. Determine the slope of the secant line and explain its relationship to the moving object.
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15. Instantaneous velocity Consider the position function sHtL = -16 t2 + 128 t (Exercise 9). Complete the following
table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous
velocity at t = 1.
Time interval
@1, 2D
@1, 1.5D
@1, 1.1D
@1, 1.01D
@1, 1.001D
Average velocity
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16. Instantaneous velocity Consider the position function sHtL = -4.9 t2 + 30 t + 20 (Exercise 10). Complete the
following table with the appropriate average velocities. Then make a conjecture about the value of the
instantaneous velocity at t = 2.
Time interval
@2, 3D
@2, 2.5D
@2, 2.1D
@2, 2.01D
@2, 2.001D
Average velocity
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17. Instantaneous velocity The following table gives the position sHtL of an object moving along a line at time t.
Determine the average velocities over the time intervals @1, 1.01D, @1, 1.001D, and @1, 1.0001D. Then make a
conjecture about the value of the instantaneous velocity at t = 1.
t
1
1.0001
1.001
1.01
sHtL 64 64.00479984 64.047984 64.4784
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18. Instantaneous velocity The following table gives the position sHtL of an object moving along a line at time t.
Determine the average velocities over the time intervals @2, 2.01D, @2, 2.001D, and @2, 2.0001D. Then make a
conjecture about the value of the instantaneous velocity at t = 2.
t
2
2.0001
2.001
2.01
sHtL 56 55.99959984 55.995984 55.9584
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19. Instantaneous velocity Consider the position function sHtL = -16 t2 + 100 t. Complete the following table with
the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t = 3.
Time interval Average velocity
@2, 3D
@2.9, 3D
@2.99, 3D
@2.999, 3D
@2.9999, 3D
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20. Instantaneous velocity Consider the position function sHtL = 3 sin t that describes a block bouncing vertically on
a spring. Complete the following table with the appropriate average velocities. Then make a conjecture about the
p
value of the instantaneous velocity at t = .
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Chapter 2 • Limits
Average velocity
Time interval
@pê2, pD
@pê2, pê2 + 0.1D
@pê2, pê2 + 0.01D
@pê2, pê2 + 0.001D
@pê2, pê2 + 0.0001D
Further Explorations
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21–24. Instantaneous velocity For the following position functions, make a table of average velocities similar to
those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.
21. sHtL = -16 t2 + 80 t + 60
at t = 3
p
22. sHtL = 20 cos t
at t =
2
23. sHtL = 40 sin 2 t
24. sHtL =
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20
at t = 0
at t = 0
t+1
25–28. Slopes of tangent lines For the following functions, make a table of slopes of secant lines and make a
conjecture about the slope of the tangent line at the indicated point.
25. f HxL = 2 x2
at x = 2
p
26. f HxL = 3 cos x
at x =
2
27. f HxL = e
x
at x = 0
28. f HxL = x3 - x
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at x = 1
29. Tangent lines with zero slope
a. Graph the function f HxL = x2 - 4 x + 3.
b. Identify the point Ha, f HaLL at which the function has a tangent line with zero slope.
c. Confirm your answer to part (b) by making a table of slopes of secant lines to approximate the slope of the
tangent line at this point.
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30. Tangent lines with zero slope
a. Graph the function f HxL = 4 - x2 .
b. Identify the point Ha, f HaLL at which the function has a tangent line with zero slope.
c. Consider the point Ha, f HaLL found in part (b). Is it true that the secant line between Ha - h, f Ha - hLL and
Ha + h, f Ha + hLL has slope zero for any value of h ∫ 0?
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31. Zero velocity A projectile is fired vertically upward and has a position given by sHtL = -16 t2 + 128 t + 192, for
0  t  9.
a. Graph the position function for 0  t  9.
b. From the graph of the position function, identify the time at which the projectile has an instantaneous
velocity of zero; call this time t = a.
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Section 2.1 The Idea of Limits
c. Confirm your answer to part (b) by making a table of average velocities to approximate the instantaneous
velocity at t = a.
d. For what values of t on the interval @0, 9D is the instantaneous velocity positive (the projectile moves
upward)?
e. For what values of t on the interval @0, 9D is the instantaneous velocity negative (the projectile moves
downward)?
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32. Impact speed A rock is dropped off the edge of a cliff and its distance s (in feet) from the top of the cliff after t
seconds is sHtL = 16 t2 . Assume the distance from the top of the cliff to the water below is 96 ft.
a. When will the rock strike the water?
b. Make a table of average velocities and approximate the velocity at which the rock strikes the water.
p
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33. Slope of tangent line Given the function f HxL = 1 - cos x and the points A
p
p
p
+ 0.05 , C
p
+ 0.5, f
2
,
2
+ 0.5 , and DHp, f HpLL (see figure), find the slopes of the secant
2
2
2
2
lines through A and D, A and C, and A and B. Use your calculations to make a conjecture about the slope of the
p
line tangent to the graph of f at x = .
2
B
+ 0.05, f
p
, f
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