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608
CHAPTER 9 THE DERIVATIVE
9.1 Concept Questions
1-2
4. What is an indeterminate form? Illustrate with an
example.
5. Explain in your own words the meaning of lim f(x) = L
and lim f(x) = M.
X-3
1. Explain what is meant by the statement lim f(x) = 3.
2. a. If lim f(x) = 5, what can you say about f(3)? Explain.
b. If f(2) = 6, what can you say about lim f(x)? Explain.
3. Evaluate the following, and state the property of limits that
you use at each step.
2x2 + x + 5
a. lim Vx(2x2 + 1) b. lim
x + 1
x
x-2
-50
3/2
1-4
9.1 Exercises
In Exercises 1-8, use the graph of the given function fto determine
lim f(x) at the indicated value of a, if it exists.
xa
In Exercises 9-16, complete the table by computing f(x) at the
given values of x. Use these results to estimate the indicated
limit (if it exists).
9. f(x) = x² + 1; lim f(x)
1. y = f(x)
y
2.
y = f(x)
3
2.
.
1.9
1.99
1.999
2.001
2.01
2.1
2
1
f(x)
.
-3 -2 -1
1 2 3
-3-2
1 2 3
-1
10. f(x) = 2x²
- 1; lim f(x)
a= -2
a=1
0.9
0.99
0.999
1.001
1.01
1.1
3.
AY
4.
f(x)
y = f(x)
4
3
1.x
3
2
11. f(x)
lim f(x)
y = f(x)
X
1-0
1
.
-0.1
-0.01
-0.001
0.001
0.01
0.1
-2 -1
1
X
2 3
a = 1
f(x)
-3 -2 -1
1
2
3
a = 3
x
12. f(x)
11
"; lim f(x)
5.
6.
x-1
y=f(x)
3+
4
x
0.9
0.99
0.999
1.001
1.01
1.1
2
3-
f(x)
y = f(x)
1+
2
1
1
-3-2-1
1
2
3
13. f(x) =
(x
lim f(x)
1)"
.
a= -2
24 -3 -2 -1
12
a = -2
.
0.9
0.99
0.999
1.001
1.01
1.1
7.
AY
8.
Y
3
y = f(x)
3
2
y = f(x)
14. f(x)
1
lim f(x)
x - 212
2
1
X
1.9
1.99
1.999
2.001
2.01
2.1
-4-3-2-1
12
-3-2-1
+4
3
1
2
-2
a = -2
a = 0
x2 + x - 2
15. f(x)
lim f(x)
X-1
.
0.9
0.99
0.999
1.001
1.01
1.1
f(x)
Unless otherwise noted, all content on this page is Cengage Learning
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapters)
Editorial review has deemed that any suppressed content does not materially affect the overall leaming experience. Cengage Leaming reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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9.1 LIMITS
609
X-1
16. f(x)
lim f(x)
45. lim Vg(x)
46. lim V 5f(x) + 3g(x)
x-1-1
x-
X-
.
0.9
0.99
0.999
1.001
1.01
1.1
$(x)
2f(x) - g(x)
47. lim
f(x)g(x)
g(x) - f(x)
48. lim
-a f(x) + Vg(x)
In Exercises 17-22, sketch the graph of the function f, and eval-
uate lim f(x), if it exists, for the given value of a.
In Exercises 49-62, find the indicated limit, if it exists.
r? - 1
x2 - 4
49. lim
50. lim
X-X-1
X-2 x + 2
x-a
17. f(x)
x-1
1-1
if x < 0
if x > 0

(a = 0)

3x

x – x

51. lim

.

2.x2

52. lim

X

18. f(x)

Sx-1 if x < 3
- 2x + 8 if x > 3

(a = 3)

x2 – 25

53. lim

—5x + 5

b + 1

54. lim

–3b + 3

X

19. f(x)

if x < 1
if x = 1
if x > 1

0

-X + 2

(a = 1)

X

55. lim

2-3-1

x + 2

56. lim

2+28 – 2

– x – 6

57. lim

*–2x + x – 2

58. lim

20. f(x)

– 8

2.

– 2x + 4 if x 1

Slxl if x = 0

(a = 0)

1 if x = 0

X-4

60. lim

3-4 VX-2

21. f(x)

Vr-1

59. lim

1-X-1

Vx+1

Hint: Multiply by

Vx+1

=

Hint: See Exercise 59.

x – 1|

22. f(x)

{

if x # 1

if x = 1

(a = 1)

X-1

61. lim

*- x + x2 – 2x

4 – x2

62. lim

-~-2 2×2 + x3

In Exercises 23-40, find the indicated limit.

23. lim 3

24. lim -3

X-2

In Exercises 63-68, use the graph of the function f to determine

lim f(x) and lim f(x), if they exist.

X2

25. lim x

26. lim – 3x

–2

63.

64.

AY

f(x) = 2×2 – 10

f(x) = x3 – x

27. lim (1 – 2x)

28. lim (412 – 2t + 1)

29. lim (2x – 3×2 + x + 2)

10

30. lim (4×5 – 20×2 + 2x + 1)

-1

1 2 3

10

3

31. lim (2s2 – 1)(2s + 4) 32. lim (x2 + 1)(x2 – 4)

3-0

x2

65.

66.

2x + 1

33. lim

*2 x + 2

x2

x3 + 1

34. lim

– 2×3 + 2

36. lim V 5x + 2

1-2

35. lim Var + 2

f(x) =

1

37. lim V2x4 + x2

38. lim

x-12

121² + 4

x? + 1

-3 -2 -1

-2

1

2

メ→-3

3

67.

AY

V x2 + 8

39. lim

*–1 2x + 4

XV x2 + 7

40. lim

**3 2x – 2x + 3

f(x) = 2 – 4

3

In Exercises 41-48, find the indicated limit given that

lim f(x) = 3 and lim g(x) = 4.

A

X

-1

41. lim (f(x) – g(x)]

42. lim 2f(x)

43. lim (2f(x) – 3g(x)]

44. lim [f(x)g(x)]

Unless otherwise noted, all content on this page is Cengage Learning

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or Chapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

2:25 PM Mon Sep 28

100%

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chegg.com
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Chegg eReader
Chegg eReader: Applied Mathematics for the Manageri...
<
Q
610
CHAPTER 9 THE DERIVATIVE
68.
y
81. Toxic WASTE A city's main well was recently found to be
contaminated with trichloroethylene, a cancer-causing chem-
ical, as a result of an abandoned chemical dump leaching
chemicals into the water. A proposal submitted to city coun-
cil members indicates that the cost, measured in millions of
dollars, of removing x% of the toxic pollutant is given by
1
V-X
SV-
if x < 0
f(x)
=
x
if x > 0

-2 – 1

1

2

3

x + 1

C(x)

0.5x

100 – x

(0

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