James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

94 Chapter 2 Limits and Derivatives
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31– 43 Determine the infinite limit.
x 1 1
31. lim
32.
x l5 1 x 2 5
33. lim
x l1
35.
37.
39.
41.
42.
2 2 x
34. lim
2
sx 2 1d x l3 2
x 1 1
lim
x l5 2 x 2 5
sx
sx 2 3d 5
lim lnsx 2 2 9d 36. lim lnssin xd
x l3 1 x l 01 1
lim sec x 38. lim
xlsy2d 1 x cot x
x l 2
x 2 2 2x
lim x csc x 40. lim
x l22 x l 2 2 x 2 2 4x 1 4
x 2 2 2x 2 8
lim
x l2 1 x 2
1S 2 5x 1 6
lim D
1 xl0 x 2 ln x
43. lim
xl0
sln x 2 2 x 22 d
44. (a) Find the vertical asymptotes of the function
y − x 2 1 1
3x 2 2x 2
(b) Confirm your answer to part (a) by graphing the
function.
1
1
45. Determine lim and lim
x l1 2 x 3 2 1 x l1 1 x 3 2 1
(a) by evaluating f sxd − 1ysx 3 2 1d for values of x that
approach 1 from the left and from the right,
(b) by reasoning as in Example 9, and
(c) from a graph of f.
46. (a) By graphing the function f sxd − stan 4xdyx and
zooming in toward the point where the graph crosses
the y-axis, estimate the value of lim x l 0 f sxd.
(b) Check your answer in part (a) by evaluating f sxd for
values of x that approach 0.
47. (a) Estimate the value of the limit lim x l 0 s1 1 xd 1yx to
five decimal places. Does this number look familiar?
(b) Illustrate part (a) by graphing the function
y − s1 1 xd 1yx .
48. (a) Graph the function f sxd − e x 1 ln| x 2 4 | for
0 < x < 5. Do you think the graph is an accurate
representation of f ?
(b) How would you get a graph that represents f better?
49. (a) Evaluate the function f sxd − x 2 2 s2 x y1000d for
x − 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the
value of
lim
x l 0Sx 2 2
1000D
2x
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(b) Evaluate f sxd for x − 0.04, 0.02, 0.01, 0.005, 0.003,
and 0.001. Guess again.
50. (a) Evaluate hsxd − stan x 2 xdyx 3 for x − 1, 0.5, 0.1,
0.05, 0.01, and 0.005.
tan x 2 x
(b) Guess the value of lim .
x l 0 x 3
(c) Evaluate hsxd for successively smaller values of x
until you finally reach a value of 0 for hsxd. Are you
still confident that your guess in part (b) is correct?
Explain why you eventually obtained 0 values. (In
Section 4.4 a method for evaluating this limit will be
explained.)
(d) Graph the function h in the viewing rectangle f21, 1g
by f0, 1g. Then zoom in toward the point where the
graph crosses the y-axis to estimate the limit of hsxd
as x approaches 0. Continue to zoom in until you
observe distortions in the graph of h. Compare with
the results of part (c).
51. Graph the function f sxd − sinsyxd of Example 4 in
the viewing rectangle f21, 1g by f21, 1g. Then zoom in
toward the origin several times. Comment on the behavior
of this function.
52. Consider the function f sxd − tan 1 x .
(a) Show that f sxd − 0 for x − 1 , 1
2 , 1
3 , . . .
(b) Show that f sxd − 1 for x − 4 , 4
5 , 4
9 , . . .
(c) What can you conclude about lim
x l 0 1 tan 1 x ?
53. Use a graph to estimate the equations of all the vertical
asymptotes of the curve
y − tans2 sin xd
2 < x <
Then find the exact equations of these asymptotes.
54. In the theory of relativity, the mass of a particle with
velocity v is
m 0
m −
s1 2 v 2 yc 2
where m 0 is the mass of the particle at rest and c is the
speed of light. What happens as v l c 2 ?
55. (a) Use numerical and graphical evidence to guess the
value of the limit
x 3 2 1
lim
xl1 sx 2 1
(b) How close to 1 does x have to be to ensure that the
fun ction in part (a) is within a distance 0.5 of its limit?
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