5128_Ch08_434-471

f ( x)
23. Same rate, because lim 1 x→ g ( x)
5
19. lim
u→0 tan uu 1 20. lim
21. lim x 3
3x2
1
x→ 2x
2
x 3
u→
u 2 sin ( 1 u )
22. lim 3 x2
x 1
x→ x
4
x
3 0
2
In Exercises 23–34, determine whether f grows faster than, slower
than, or at the same rate as g as x→. Give reasons for your answer.
23. f x x, gx 5x 24. f x log 2 x, gx log 3 x
25. f x x, gx x 1 x 26. f x x
,
100
gx xe x
27. f x x, gx tan 1 x 28. f x csc 1 x, gx 1 x
29. f x x ln x , gx x log 2 x 30. f x 3 x , gx 2 x
31. f x ln 2x, gx ln x 2 f ( x)
Same rate, because lim 1 x→ g ( x)
2
32. f x 10x 3 2x 2 , gx e x f ( x)
Slower, because lim 0
x→ g ( x)
33. f x tan 1 1 x , gx 1 x f ( x)
Same rate, because lim 1
x→ g ( x)
34. f x sin 1 1 x , gx 1
f ( x)
x2
Faster, because lim
x→ g ( x)
In Exercises 35 and 36,
(a) show that f has a removable discontinuity at x 0.
(b) define f at x 0 so that it is continuous there.
35. f x 2 sin x
1
e
x
1
36. f x x ln x
In Exercises 37–48, evaluate the improper integral or state that it
diverges.
37. dx
1 x
3/2 2 38. dx
1 x 2 ln (54)
7x
12
1
3
3dx
dx
39. 3x x 2 2 ln (2) 40.
0 9
x 2
1
41. 0
0
43. 2
45.
0
1
ln(x) dx –1 42. 1
du
(u 1) 3/5 0
dy
y
2/3 6
44. 2dx
x
2
ln (3)
2x
0
x 2 e x dx 2 46. xe 3x dx
24. Same rate, because lim f ( x)
l n
3
x→ g(
x)
ln
2
3
19
47. dx
e x
e x p/2 48. 4dx
x
2
p
16
In Exercises 49 and 50, use the comparison test to determine whether
the improper integral converges or diverges.
49.
ln z
dz diverges 50. e
t
dt converges
1 z
1 t
51. The second and fifth terms of a geometric sequence are 3 and
3/8, respectively. Find (a) the first term, (b) the common ratio,
and (c) an explicit formula for the nth term. (a) 6 (b) 12
(c) a n 3(2 2n )
f ( x)
f ( x)
25. Same rate, because lim 1 27. Faster, because lim
x→ g ( x)
x→ g ( x)
f ( x)
f ( x)
26. Faster, because lim 28. Same rate, because lim 1
x→ g ( x)
x→ g ( x)
Chapter 8 Review Exercises 471
52. (a) 13 (b) 1.5 (c) a n 1.5n 14.5
52. The second and sixth terms of an arithmetric sequence are 11.5
and 5.5, respectively. Find (a) the first term, (b) the common
difference, and (c) an explicit formula for the nth term.
53. Consider the improper e2⏐x⏐ dx.
(a) Express the improper integral as a limit of definite integrals.
(b) Evaluate the integral. (a) lim
b→ 0 e 2x dx lim
b
b→
b
e 2x dx (b) 1
0
54. Infinite Solid The infinite region bounded by the coordinate
axes and the curve y ln x in the first quadrant (see figure) is
revolved about the x-axis to generate a solid. Find the volume of
the solid. 2p
[0, 2] by [–1, 5]
55. Infinite Region Find the area of the region in the first
quadrant under the curve y xe x (see figure). 1
[0, 5] by [–0.5, 1]
AP* Examination Preparation
You should solve the following problems without using
a graphing calculator.
56. Consider the infinite region R in the first quadrant under the
curve y xe x/2 .
(a) Write the area of R as an improper integral.
(b) Express the integral in part (a) as a limit of a definite integral.
(c) Find the area of R.
57. The infinite region in the first quadrant bounded by the coordinate
axes and the curve y 1 1 is revolved about the y-axis to
x
generate a solid.
(a) Write the volume of the solid as an improper integral.
(b) Express the integral in part (a) as a limit of a definite integral.
(c) Find the volume of the solid.
58. Determine whether or not
0 xex dx converges. If it converges,
give its value. Show your reasoning.
f ( x)
29. Slower, because lim 0
x→∞ g ( x)
f ( x)
30. Slower, because lim 0
x→∞ g ( x)