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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan
100 Chapter 1 • Limits and Continuity
8. 2
9. False
10. True
11. 14
12. –3
13. 0
14. 18
15. 1
16. 1
17. 6
18. 1 2
19. 11
20. 10
21. 2 78
22. 0
23. 7+
3
24. 2
25. 0
26. 0
27. 5
28. 1
29. − 31
8
30. –3
31. 10
32. 14 3
33. 13 4
34. 1 5
35. 4
8. If the domain of a rational function R is {x | x = 0},
then lim R(x) = R( ).
x→2
9. True or False Properties of limits cannot be used for one-sided
limits.
(x + 1)(x + 2)
10. True or False If f (x) = and g(x) = x + 2,
x + 1
then lim f (x) = lim g(x).
x→−1 x→−1
Skill Building
In Problems 11–44, find each limit using properties of limits.
11. lim [2(x + 4)] 12. lim [3(x + 1)]
x→3 x→−2
PAGE
93 13. lim [x(3x−1)(x + 2)]
x→−2
14. lim [x(x − 1)(x + 10)]
x→−1
PAGE
94 15. lim (3t − 2) 3
t→1
16. lim (−3x + 1) 2
x→0
17. lim (3 √ ( ) 1
x) 18. lim
3√ x x→4 x→8 4
√
PAGE
95 19. lim 5x − 4
x→3
20. lim
t→2
√
3t + 4
21. lim t→2
[t √ (5t + 3)(t + 4)] 22. lim
t→−1 [t 3√ (t + 1)(2t − 1)]
PAGE
95 23. lim ( √ x + x + 4) 1/2
x→3
24. lim t→2
(t √ 2t + 4) 1/3
25. lim
t→−1 [4t(t + 1)]2/3 26. lim x→0
(x 2 − 2x) 3/5
27. lim t→1
(3t 2 − 2t + 4) 28. lim x→0
(−3x 4 + 2x + 1)
PAGE
96 29. lim (2x 4 − 8x 3 + 4x − 5) 30. lim (27x 3 + 9x + 1)
x→ 1 2
x→− 1 3
PAGE
97 31.
x 2 + 4
lim √ x→4 x
32.
x 2 + 5
lim √ x→3 3x
PAGE
97 33.
2x 3 + 5x
lim
x→−2 3x − 2
PAGE
97 35.
x 2 − 4
lim x→2 x − 2
x 3 − x
37. lim
x→−1 x + 1
39. lim
x→−8
( 2x
x + 8 + 16
x + 8
√ √
PAGE
x − 2
98 41. lim x→2 x − 2
43. lim x→4
√
x + 5 − 3
(x − 4)(x + 1)
)
34. lim x→1
2x 4 − 1
3x 3 + 2
x + 2
36. lim
x→−2 x 2 − 4
x 3 + x 2
38. lim
x→−1 x 2 − 1
40. lim x→2
( 3x
x − 2 − 6
x − 2
42. lim x→3
√ x −
√
3
x − 3
44. lim x→3
√
x + 1 − 2
x(x − 3)
In Problems 45–50, find each one-sided limit using properties of limits.
45. lim
x→3 −(x2 − 4) 46. lim
x→2 +(3x2 + x)
x 2 − 9
47. lim
x→3 − x − 3
x 2 − 9
48. lim
x→3 + x − 3
√
√
49. lim
x→3 −( 9 − x 2 + x) 2 50. lim
x→2 +(2 x 2 − 4 + 3x)
)
In Problems 51–58, use the information below to find each limit.
lim f (x) = 5
x→c
lim g(x) = 2
x→c
lim h(x) = 0
x→c
51. lim x→c
[ f (x) − 3g(x)] 52. lim x→c
[5 f (x)]
53. lim [g(x)] 3 f (x)
54. lim
x→c x→c g(x) − h(x)
55. lim x→c
h(x)
g(x)
[ 1
57. lim x→c g(x)
56. lim x→c
[4 f (x) · g(x)]
] 2
58. lim x→c
3 √ 5g(x) − 3
In Problems 59 and 60, use the graphs of the functions and properties
of limits to find each limit, if it exists. If the limit does not exist, write,
“the limit does not exist,” and explain why.
59. (a) lim [ f (x) + g(x)]
x→4
y
y h(x)
10 y f (x)
(b) lim { f (x) [g(x) − h(x)]}
x→4 8
(c) lim[ f (x) · g(x)]
x→4 6
(4, 6)
(d) lim[2h(x)]
x→4
g(x)
2
(e) lim x→4 f (x)
(4, 0)
(f)
8 x
h(x)
2
lim
y g(x)
x→4
(3, 2)
f (x)
60. (a) lim x→3
{2 [ f (x) + h(x)]}
(b)
lim + h(x)]
x→3−[g(x) (c) lim x→3
3 √ h(x)
(d) lim x→3
f (x)
h(x)
(e) lim x→3
[h(x)] 3
(f)
lim[ f (x) − 2h(x)] 3/2
x→3
y
6
4
2
2
y f (x)
(3, 2)
y g(x)
3
(3, 6)
y h(x)
In Problems 61–66, for each function f, find the limit as x approaches c
of the average rate of change of f from c to x. That is, find
f (x) − f (c)
lim
x→c x − c
61. f (x) = 3x 2 , c = 1 62. f (x) = 8x 3 , c = 2
PAGE
98 63. f (x) =−2x 2 + 4, c = 1 64. f (x) = 20 − 0.8x 2 , c = 3
65. f (x) = √ x, c = 1 66. f (x) = √ 2x, c = 5
In Problems 67–72, find the limit of the difference quotient for each
f (x + h) − f (x)
function f . That is, find lim
.
h→0 h
67. f (x) = 4x − 3 68. f (x) = 3x + 5
69. f (x) = 3x 2 + 4x + 1 70. f (x) = 2x 2 + x
PAGE
99 71. f (x) = 2 x
72. f (x) = 3 x 2
x
36. − 1 4
37. 2
38. − 1 2
39. 2
40. 3
2
41.
4
3
42.
6
1
43.
30
1
44.
12
45. 5
46. 14
47. 6
48. 6
49. 9
50. 6
51. –1
52. 25
53. 8
54. 5 2
55. 0
56. 40
57. 1 4
3
58. 7
59. (a) 6
(b) –16
(c) –16
(d) 0
(e) − 1 4
(f) 0
60. (a) –4
(b) 4
3
(c) − 2
(d) 0
(e) –8
(f) 8
61. 6
62. 96
63. –4
64. –4.8
65. 1 2
1
66.
10
67. 4
68. 3
69. 6x + 4
70. 4x + 1
2
71. −
x
2
6
72. −
x
3
100
Chapter 1 • Limits and Continuity
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