Odd01

C H A P T E R 1 Limits and Their Properties Section 1.1

A Preview of Calculus . . . . . . . . . . . . . . . . . . . . 27

Section 1.2

Finding Limits Graphically and Numerically . . . . . . . . 27

Section 1.3

Evaluating Limits Analytically

Section 1.4

Continuity and One-Sided Limits . . . . . . . . . . . . . . 37

Section 1.5

Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . 42

. . . . . . . . . . . . . . . 31

Review Exercises

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Problem Solving

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

C H A P T E R 1 Limits and Their Properties Section 1.1

A Preview of Calculus

Solutions to Odd-Numbered Exercises 1. Precalculus: 20 ft sec 15 seconds 300 feet

3. Calculus required: slope of tangent line at x 2 is rate of change, and equals about 0.16.

5. Precalculus: Area 12 bh 12 5 3 15 2 sq. units

7. Precalculus: Volume 2 4 3 24 cubic units

9. (a)

6

(1, 3) −4

8 −2

(b) The graphs of y2 are approximations to the tangent line to y1 at x 1. (c) The slope is approximately 2. For a better approximation make the list numbers smaller:

0.2, 0.1, 0.01, 0.001 11. (a) D1 5 1 2 1 5 2 16 16 5.66 5 5 5 5 5 5 (b) D2 1 2 1 2 3 1 3 4 1 4 1 2

2

2

2

2.693 1.302 1.083 1.031 6.11 (c) Increase the number of line segments.

Section 1.2 1.

x

1.9

1.99

1.999

2.001

2.01

2.1

f x

0.3448

0.3344

0.3334

0.3332

0.3322

0.3226

lim

x→2

3.

Finding Limits Graphically and Numerically

x 2 0.3333 x2 x 2

0.1

x f x lim

x→0

0.01

0.2911

0.001

0.2889

x 3 3

x

Actual limit is 13 .

0.2887

0.2887

0.001

0.01

0.1

0.2887

0.2884

0.2863

Actual limit is 1 2 3 .

27

28

Chapter 1

5.

x

2.9

f x

0.0641

lim

x→3

7.

Limits and Their Properties

2.999

3.001

3.01

3.1

0.0627

0.0625

0.0625

0.0623

0.0610

1 x 1 1 4 0.0625 x 3

0.1

x

2.99

f x

0.01

0.9983

lim

x→0

0.001

0.99998

sin x 1.0000 x

Actual limit is 161 .

1.0000

0.001

0.01

0.1

1.0000

0.99998

0.9983

(Actual limit is 1.) (Make sure you use radian mode.)

11. lim f x lim 4 x 2

9. lim 4 x 1

x→2

x→3

13. lim

x→2

x 5 does not exist. For values of x to the left of 5, x 5 x 5 equals 1,

x 5 whereas for values of x to the right of 5, x 5 x 5 equals 1. x→5

15. lim tan x does not exist since the function increases and x→ 2

17. lim cos 1 x does not exist since the function oscillates x→0

between 1 and 1 as x approaches 0.

decreases without bound as x approaches 2.

19. C t 0.75 0.50 t 1

(a)

(b)

3

t

3

3.3

3.4

3.5

3.6

3.7

4

C 1.75

2.25

2.25

2.25

2.25

2.25

2.25

2.5

2.9

3

3.1

3.5

4

1.75

1.75

1.75

2.25

2.25

2.25

lim C t 2.25

t→3.5

(c)

5

0 0

t

2

C 1.25

lim C t does not exist. The values of C jump from 1.75 to 2.25 at t 3. t→3

21. You need to find such that 0 < x 1 < implies 1 f x 1 1 < 0.1. That is, x

0.1 <

1 1 < 0.1 x

1 0.1 <

1 x

< 1 0.1

9 < 10

1 x

11 < 10

10 > 9

x

>

10 11

10 10 1 > x 1 > 1 9 11 1 1 > x 1 > .

9 11

So take

1 . Then 0 < x 1 < implies 11

1 1 < x 1 < 11 11

1 1 < x 1 < .

11 9

Using the first series of equivalent inequalities, you obtain

f x 1

1 1 < < 0.1. x

Section 1.2 23. lim 3x 2 8 L x→2

3x 2 8 3x 6 3 x 2

< 0.01 < 0.01 < 0.01

0.01 0.0033 3 0.01 Hence, if 0 < x 2 < , you have 3

0 < x 2 <

3x 6 < 0.01 3x 2 8 < 0.01 f x L < 0.01 3 x 2 < 0.01

27. lim x 3 5

Finding Limits Graphically and Numerically

25. lim x2 3 1 L x→2

x2 3 1 x2 4 x 2 x 2 x 2 x 2

< 0.01 < 0.01

If we assume 1 < x < 3, then 0.01 5 0.002.

Hence, if 0 < x 2 < 0.002, you have 1

x 2 x 2 < 0.01 x2 4 < 0.01 x2 3 1 < 0.01 f x L < 0.01

12 x 1 12 4 1 3

x 1 3 < x 2 <

<

1 2

<

Hence, let .

1 2

1 2

Hence, if 0 < x 2 < , you have

x 2 <

x 3 5 < f x L <

x 4 x 4

< < 2

Hence, let 2 .

Hence, if 0 < x 4 < 2 , you have

x 4 < 2

x 2 x 1 3 1 2

1 2

< <

f x L <

31. lim 3 3

3 33. lim x 0

x→6

x→0

Given > 0:

x <

3 x 0 < Given > 0:

3 3 <

3

x

0 <

< 3

Hence, any > 0 will work.

Hence, let 3.

Hence, for any > 0, you have

Hence for 0 < x 0 < 3, you have

1

x 2 < 0.002 5 0.01 < x 2 0.01

Given > 0:

x 3 5 x 2

< 0.01

0.01

x→ 4

Given > 0:

< 0.01

x 2 < x 2

29. lim

x→2

f x L < 3 3 <

29

x < < 3 x 0 < f x L <

3

3 x

30

Chapter 1

Limits and Their Properties

35. lim x 2 2 2 4 x→ 2

37. lim x2 1 2 x→1

Given > 0:

Given > 0:

x 2 4 x 2 4 x 2 x 2 x 2

x2 1 2 x2 1 x 1 x 1

<

x 2 < 0

< <

Hence, .

x 2 < x 2 4 < x 2 4 < f x L <

If we assume 0 < x < 2, then 3.

Hence for 0 < x 1 < , you have 3

1

1

x2 1 < x2 1 2 < f x 2 < x 9

0.5

41. f x

x 4 1 6

x 1 < 3 < x 1

(because x 20)

x 5 3

lim f x

<

x 2 <

x→4

<

x 1 < x 1

Hence for 0 < x 2 < , you have

39. f x

<

−6

10

x 3

lim f x 6

6

x→9

−0.1667

0

10 0

The domain is 5, 4 4, .

1 The graphing utility does not show the hole at 4, 6 .

The domain is all x ≥ 0 except x 9. The graphing utility does not show the hole at 9, 6 .

43. lim f x 25 means that the values of f approach 25 as x gets closer and closer to 8. x→8

45. (i) The values of f approach different numbers as x approaches c from different sides of c:

(ii) The values of f increase without bound as x approaches c:

(iii) The values of f oscillate between two fixed numbers as x approaches c:

y

y

y

6

4

5

3

4

2

3

4 3

2

1

1

x

−4 −3 −2 −1 −1

1

2

3

4

−3 −2 −1 −1

3

4

−3

−4

−4

47. f x 1 x 1 x lim 1 x 1 x e 2.71828

x→0

y 7

3

(0, 2.7183)

2 1 −3 −2 −1 −1

x 1

2

2

5

−2

−3

x

−4 −3 −2

x 2

3

4

5

x

f x

x

f x

0.1

2.867972

0.1

2.593742

0.01

2.731999

0.01

2.704814

0.001

2.719642

0.001

2.716942

0.0001

2.718418

0.0001

2.718146

0.00001

2.718295

0.00001

2.718268

0.000001

2.718283

0.000001

2.718280

3

4

Section 1.3 49. False; f x sin x x is undefined when x 0.

From Exercise 7, we have lim

x→0

51. False; let f x

sin x 1. x

Evaluating Limits Analytically

31

53. Answers will vary.

x10, 4x, 2

x 4 . x 4

f 4 10 lim f x lim x2 4x 0 10

x→4

x→4

55. If lim f x L1 and lim f x L2, then for every > 0, there exists 1 > 0 and 2 > 0 such that x c < 1 ⇒ f x L1 < and x→c

x→c

x c < 2 ⇒ f x L2 < . Let equal the smaller of 1 and 2. Then for x c L1 L2 L1 f x f x L2 ≤ L1 f x f x L2 < .

Therefore, L1 L2 < 2 . Since > 0 is arbitrary, it follows that L1 L2.

< , we have

57. lim f x L 0 means that for every > 0 there exists > 0 such that if x→c

0 < x c < , then

f x L 0

< .

This means the same as f x L < when 0 < x c < .

Thus, lim f x L. x→c

Section 1.3 1.

Evaluating Limits Analytically (a) lim h x 0

7

x→5

−8

3.

(b) lim h x 6

x→0

π

−π

x→ 1

13

(a) lim f x 0

4

−7

(b) lim f x 0.524

−4

h x x2 5x

f x x cos x 7. lim 2x 1 2 0 1 1

5. lim x4 24 16 x→2

x→0

9. lim x2 3x 3 2 3 3 9 9 0 x→ 3

11. lim 2x2 4x 1 2 3 2 4 3 1 18 12 1 7 x→ 3

13. lim

x→2

17. lim

x→7

1 1

x 2 5x

x 2

15. lim

x→1

5 7

7 2

35

9

35 3

x 3 1 3 2 2

x2 4 12 4 5 5

19. lim x 1 3 1 2 x→3

x→ 3

6

32

Chapter 1

Limits and Their Properties

21. lim x 3 2 4 3 2 1

23. (a) lim f x 5 1 4

x→ 4

x→1

(b) lim g x 43 64 x→4

(c) lim g f x g f 1 g 4 64 x→1

25. (a) lim f x 4 1 3

27. lim sin x sin x→ 2

x→1

(b) lim g x 3 1 2

1 2

x→3

(c) lim g f x g 3 2 x→1

29. lim cos x→2

33.

x 2 1 cos

3 3 2

lim sin x sin

x→5 6

31. lim sec 2x sec 0 1 x→0

5 1

6 2

35. lim tan x→3

37. (a) lim 5g x 5 lim g x 5 3 15 x→c

x→c

(b) lim f x g x lim f x lim g x 2 3 5 x→c

x→c

x→c

(c) lim f x g x lim f x lim g x 2 3 6 x→c

x→c

x→c

lim f x

f x 2 x→c (d) lim

x→c g x lim g x 3

39. (a) lim f x 3 lim f x 3 4 3 64 x→c

x→c

(b) lim f x lim f x 4 2 x→c

x→c

(c) lim 3 f x 3 lim f x 3 4 12 x→c

x→c

(d) lim f x 3 2 lim f x 3 2 4 3 2 8 x→c

x→c

41. f x 2x 1 and g x x 0.

4x tan 34 1

2x2 x agree except at x

x→c

43. f x x x 1 and g x

x3 x agree except at x 1. x 1

(a) lim g x lim f x 1

(a) lim g x lim f x 2

(b) lim g x lim f x 3

(b) lim g x lim f x 0

x→0

x→ 1

45. f x

x→1

x→0

x→ 1

x→ 1

x2 1 and g x x 1 agree except at x 1. x 1

lim f x lim g x 2

x→ 1

x→1

47. f x x 2.

x→ 1

x→ 1

x3 8 and g x x2 2x 4 agree except at x 2

lim f x lim g x 12

x→2

3

x→2

12 −3

4

−4

−9

9 0

49. lim

x→5

x 5 x 5 lim x2 25 x→5 x 5 x 5 lim

x→5

1 1

x 5 10

51. lim

x→ 3

x2 x 6 x 3 x 2 lim x→ 3 x 3 x 3 x2 9 lim

x→ 3

x 2 5 5

x 3 6 6

Section 1.3

53. lim

x 5 5

lim

x

x→0

lim

x→0

55. lim

x 5 3

x 4

x→4

x 5 5

lim

x→4

x 5 5

x 5 5

5 x 5 5 1 1 lim

x x 5 5 x→0 x 5 5 2 5 10

x 5 3

x 4

x→4

lim

x

x→0

Evaluating Limits Analytically

x 5 3

x 5 3

1 x 5 9 1 1

lim

x 4 x 5 3 x→4 x 5 3 9 3 6

1 1 2 2 x

1 1 2 x 2 2 2 x 57. lim lim lim

x→0 x→0 x→0 2 2 x x x 4

59. lim

x→0

2 x x 2x 2x 2 x 2x lim lim 2 2

x→0

x→0

x

x

x x 2 2 x x 1 x2 2x 1 x2 2x x x 2 2x 2 x 1 x2 2x 1 lim

x→0

x→0

x

x

61. lim

lim 2x x 2 2x 2

x→0

63. lim

x 2 2

x→0

x

x

0.1

f x

0.354 0.01

0.358

2

0.001

0

0.001

0.01

0.1

0.345

?

0.354

0.353

0.349

0.354

−3

3

−2

Analytically, lim

x 2 2

x

x→0

lim

x 2 2

x

x→0

lim

x→0

x 2 2

x 2 2

x 2 2

x x 2 2

lim

x→0

1

x 2 2

1 2 2

2

4

1 1

2 x 2 1 65. lim

x→0 x 4

0.354

3

−5

x

0.1

0.01

0.001

0

0.001

f x

0.263

0.251

0.250

?

0.250

1 1

2 x 2 2 2 x Analytically, lim lim x→0 x→0 x 2 2 x

1

0.01

0.1

0.249

0.238

x

1

1

−2

1

1

lim . x x→0 lim 2 2 x x x→0 2 2 x 4

33

34

Chapter 1

67. lim

xâ&#x2020;&#x2019;0

Limits and Their Properties

sin x lim xâ&#x2020;&#x2019;0 5x

sin x x

15 1 15 51

69. lim

xâ&#x2020;&#x2019;0

1 sin x 1 cos x lim xâ&#x2020;&#x2019;0 2 2x2

sin x x

1 cos x x

1 1 0 0 2

1 cos h 2 1 cos h 1 cos h lim hâ&#x2020;&#x2019;0 hâ&#x2020;&#x2019;0 h h

sin2 x sin x lim sin x 1 sin 0 0 xâ&#x2020;&#x2019;0 xâ&#x2020;&#x2019;0 x x

71. lim

73. lim

0 0 0

75. lim

xâ&#x2020;&#x2019; 2

cos x lim sin x 1 xâ&#x2020;&#x2019; 2 cot x

79. f t

f t

tâ&#x2020;&#x2019;0

sin 3t sin 3t lim tâ&#x2020;&#x2019;0 2t 3t

4

0.01

2.96

0.001

2.9996

3

0

0.001

0.01

0.1

?

3

2.9996

2.96

â&#x2C6;&#x2019; 2

2 â&#x2C6;&#x2019;1

The limit appear to equal 3.

sin 3t sin 3t lim 3 3 1 3. tâ&#x2020;&#x2019;0 tâ&#x2020;&#x2019;0 t 3t

Analytically, lim

81. f x

32 1 32 23

sin 3t t 0.1

t

77. lim

sin x2 x

1

â&#x2C6;&#x2019; 2

x

0.1

0.01 0.001 0 0.001 0.01 0.1

f x

0.099998 0.01 0.001 ? 0.001 0.01 0.099998

2

â&#x2C6;&#x2019;1

sin x2 sin x2 lim x 0 1 0. xâ&#x2020;&#x2019;0 xâ&#x2020;&#x2019;0 x x2

Analytically, lim

83. lim

hâ&#x2020;&#x2019;0

f x h f x 2 x h 3 2x 3 2x 2h 3 2x 3 2h lim lim lim 2 hâ&#x2020;&#x2019;0 hâ&#x2020;&#x2019;0 hâ&#x2020;&#x2019;0 h h h h

4 4

4 4 f x h f x 4x 4 x h x h x 85. lim lim lim lim 2 hâ&#x2020;&#x2019;0 x h x hâ&#x2020;&#x2019;0 hâ&#x2020;&#x2019;0 hâ&#x2020;&#x2019;0 h h x h xh x 87. lim 4 x2 â&#x2030;¤ lim f x â&#x2030;¤ lim 4 x2 xâ&#x2020;&#x2019;0

xâ&#x2020;&#x2019;0

xâ&#x2020;&#x2019;0

89. f x x cos x 4

4 â&#x2030;¤ lim f x â&#x2030;¤ 4 xâ&#x2020;&#x2019;0

Therefore, lim f x 4. xâ&#x2020;&#x2019;0

â&#x2C6;&#x2019; 3 2

3 2

â&#x2C6;&#x2019;4

lim x cos x 0

xâ&#x2020;&#x2019;0

Section 1.3

91. f x x sin x

93. f x x sin

6

2

1 x

â&#x2C6;&#x2019;0.5

0.5

â&#x2C6;&#x2019;6

â&#x2C6;&#x2019;0.5

lim x sin x 0

lim x sin

xâ&#x2020;&#x2019;0

95. We say that two functions f and g agree at all but one point (on an open interval) if f x g x for all x in the interval except for x c, where c is in the interval.

1 0 x

97. An indeterminant form is obtained when evaluating a limit using direct substitution produces a meaningless fractional expression such as 0 0. That is, lim

xâ&#x2020;&#x2019;c

f x g x

for which lim f x lim g x 0 xâ&#x2020;&#x2019;c

99. f x x, g x sin x, h x

When you are â&#x20AC;&#x153;close toâ&#x20AC;? 0 the magnitude of f is approximately equal to the magnitude of g.

Thus, g f 1 when x is â&#x20AC;&#x153;close toâ&#x20AC;? 0.

f h

â&#x2C6;&#x2019;5

xâ&#x2020;&#x2019;c

sin x x

3

g

5

â&#x2C6;&#x2019;3

101. s t 16t2 1000 lim tâ&#x2020;&#x2019;5

s 5 s t 600 16t2 1000 16 t 5 t 5 lim lim lim 16 t 5 160 ft sec. tâ&#x2020;&#x2019;5 tâ&#x2020;&#x2019;5 tâ&#x2020;&#x2019;5 5 t 5 t t 5

Speed 160 ft sec 103. s t 4.9t2 150 s 3 s t 4.9 32 150 4.9t2 150 4.9 9 t2 lim lim tâ&#x2020;&#x2019;3 tâ&#x2020;&#x2019;3 tâ&#x2020;&#x2019;3 3 t 3 t 3 t

lim

lim

xâ&#x2020;&#x2019;3

4.9 3 t 3 t lim 4.9 3 t 29.4 m sec xâ&#x2020;&#x2019;3 3 t

105. Let f x 1 x and g x 1 x. lim f x and lim g x do not exist. xâ&#x2020;&#x2019;0

xâ&#x2020;&#x2019;0

lim 0 0

1 1

lim f x g x lim xâ&#x2020;&#x2019;0 xâ&#x2020;&#x2019;0 x x

xâ&#x2020;&#x2019;0

107. Given f x b, show that for every > 0 there exists a > 0 such that f x b < whenever x c < . Since f x b b b 0 < for any > 0, then any value of > 0 will work.

35

0.5

â&#x2C6;&#x2019; 2

xâ&#x2020;&#x2019;0

Evaluating Limits Analytically

109. If b 0, then the property is true because both sides are equal to 0. If b 0, let > 0 be given. Since lim f x L,

xâ&#x2020;&#x2019;c

there exists > 0 such that f x L < b whenever 0 < x c < . Hence, wherever 0 < x c < , we have

b f x L

<

or

bf x bL

which implies that lim bf x bL. xâ&#x2020;&#x2019;c

<

36

Chapter 1

Limits and Their Properties

M f x ≤ f x g x ≤ M f x

111.

113. False. As x approaches 0 from the left,

lim M f x ≤ lim f x g x ≤ lim M f x

x→c

x→c

x→c

x 1. x

2

M 0 ≤ lim f x g x ≤ M 0 x→c

−3

0 ≤ lim f x g x ≤ 0

3

x→c

Therefore, lim f x g x 0.

−2

x →c

115. True.

117. False. The limit does not exist.

4

−3

6

−2

119. Let f x

4,4,

if x ≥ 0 if x < 0

lim f x lim 4 4.

x→0

x→0

lim f x does not exist since for x < 0, f x 4 and for x ≥ 0, f x 4.

x→0

rational 0,1, ifif xx isis irrational 0, if x is rational g x x, if x is irrational

121. f x

lim f x does not exist.

x→0

No matter how “close to” 0 x is, there are still an infinite number of rational and irrational numbers so that lim f x does not x→0 exist. lim g x 0.

x→0

When x is “close to” 0, both parts of the function are “close to” 0.

123. (a) lim

x→0

1 cos x 1 cos x lim x→0 x2 x2 lim

x→0

1 cos2 x x 1 cos x 2

sin2 x x→0 x2

lim 1

1 cos x

1 cos x

1

1 cos x

12 21

(b) Thus,

1 cos x 1 1 ⇒ 1 cos x x2 x2 2 2 1 ⇒ cos x 1 x2 for x 0.

2

1 (c) cos 0.1 1 0.1 2 0.995 2 (d) cos 0.1 0.9950, which agrees with part (c).

Section 1.4

Section 1.4

3. (a) lim f x 0

(b) lim f x 1

(b) lim f x 0

x→3

x→4

x→3

The function is continuous at x 3.

x→0

x→4

(c) lim f x does not exist

(c) lim f x 0

x→3

x

lim

x→0

9.

does not exist because

x x2 9

grows

x

1

1

lim x x→0

x x x x

x→0

x→3

x x2 9

x 1. x

lim

x→3

lim

x→ 3

without bound as x → 3 .

1 1

x x x x x x lim 13. lim x→0 x→0 x x x x

15. lim f x lim

The function is NOT continuous at x 4.

The function is NOT continuous at x 3.

x 5 1 1 lim

x2 25 x→5 x 5 10

x

lim f x 2

x→4

(b) lim f x 2

x→3

(c) lim f x 1

11. lim

5. (a)

x→3

x→3

x→5

37

Continuity and One-Sided Limits

1. (a) lim f x 1

7. lim

Continuity and One-Sided Limits

1 x x x

1 1 2 x x 0 x

x 2 5

2 2

17. lim f x lim x 1 2 x→1

x→1

lim f x lim x3 1 2

x→1

x→1

lim f x 2

x→1

21. lim 3 x 5 3 3 5 4

19. lim cot x does not exist since x→

x→4

x 3 for 3 < x < 4

lim cot x and lim cot x do not exist.

x→

x→

23. lim 2 x does not exist x→3

because lim 2 x 2 3 5

x→3

and

25. f x

1 x2 4

27. f x

has discontinuities at x 2 and x 2 since f 2 and f 2 are not defined.

x x 2

has discontinuities at each integer k since lim f x lim f x . x→k

x→k

lim 2 x 2 4 6.

x→3

29. g x 25 x2 is continuous on 5, 5 .

31. lim f x 3 lim f x . x→0

x→0

f is continuous on 1, 4 .

33. f x x2 2x 1 is continuous for all real x.

38

Chapter 1

Limits and Their Properties

35. f x 3x cos x is continuous for all real x.

37. f x

x is not continuous at x 0, 1. Since x2 x

x 1 for x 0, x 0 is a removable

x2 x x 1 discontinuity, whereas x 1 is a nonremovable discontinuity.

39. f x

x is continuous for all real x. x2 1

41. f x

x 2 x 2 x 5

has a nonremovable discontinuity at x 5 since lim f x x→5 does not exist, and has a removable discontinuity at x 2 since lim f x lim

x→ 2

43. f x

x 2 has a nonremovable discontinuity at x 2 since

45. f x

x,x ,

x 2

x→ 2

1 1 . x 5 7

lim f x does not exist.

x→ 2

x ≤ 1 x > 1

2

has a possible discontinuity at x 1.

1. f 1 1 2.

lim f x lim x 1

x→1

x→1

x→1

x→1

lim f x 1

lim f x lim x2 1

x→1

3. f 1 lim f x x→1

f is continuous at x 1, therefore, f is continuous for all real x. x 1, 47. f x 2 3 x,

x ≤ 2

1. f 2

x > 2

2 1 2 2

lim f x lim

x→2

2.

has a possible discontinuity at x 2.

x→2

2x 1 2

lim f x lim 3 x 1

x→2

x→2

lim f x does not exist.

x→2

Therefore, f has a nonremovable discontinuity at x 2.

49. f x

x tan 4 , x,

x x

1. f 1 1 2. lim f x 1 x→ 1

3. f 1 lim f x x→ 1

x < 1 tan 4 ,

≥ 1 x,

1 < x < 1 has possible discontinuities at x 1, x 1. x ≤ 1 or x ≥ 1

f 1 1 lim f x 1

x→1

f 1 lim f x x→1

f is continuous at x ± 1, therefore, f is continuous for all real x.

Section 1.4

Continuity and One-Sided Limits

39

51. f x csc 2x has nonremovable discontinuities at integer multiples of 2.

53. f x x 1 has nonremovable discontinuities at each integer k.

55. lim f x 0

57. f 2 8

50

xâ&#x2020;&#x2019;0

lim f x 0

Find a so that lim ax2 8 â&#x2021;&#x2019; a

xâ&#x2020;&#x2019;0

xâ&#x2020;&#x2019;2

f is not continuous at x 2.

â&#x2C6;&#x2019;8

8 2.

22

8 â&#x2C6;&#x2019;10

59. Find a and b such that lim ax b a b 2 and lim ax b 3a b 2. xâ&#x2020;&#x2019; 1

xâ&#x2020;&#x2019;3

a b 2

3a b 2 4

4a

a 1 b

2, f x x 1, 2,

x â&#x2030;¤ 1 1 < x < 3 x â&#x2030;Ľ 3

2 1 1

61. f g x x 1 2

63. f g x

Continuous for all real x.

Nonremovable discontinuities at x Âą 1

67. f x

65. y x x Nonremovable discontinuity at each integer 0.5

1 1

x2 5 6 x2 1

2xx 2x,4, 2

x â&#x2030;¤ 3 x > 3

Nonremovable discontinuity at x 3 5

â&#x2C6;&#x2019;3

3

â&#x2C6;&#x2019;5

7

â&#x2C6;&#x2019;1.5 â&#x2C6;&#x2019;5

69. f x

x x2 1

71. f x sec

Continuous on ,

73. f x

sin x x

Continuous on: . . . , 6, 2 , 2, 2 , 2, 6 , 6, 10 , . . .

1 75. f x 16x4 x3 3 is continuous on 1, 2 .

f 1 33 16 and f 2 4. By the Intermediate Value Theorem, f c 0 for at least one value of c between 1 and 2.

3

â&#x2C6;&#x2019;4

x 4

4

â&#x2C6;&#x2019;2

The graph appears to be continuous on the interval 4, 4 . Since f 0 is not defined, we know that f has a discontinuity at x 0. This discontinuity is removable so it does not show up on the graph.

40

Chapter 1

Limits and Their Properties

77. f x x2 2 cos x is continuous on 0, . f 0 3 and f 2 1 > 0. By the Intermediate Value Theorem, f c 0 for the least one value of c between 0 and .

81. g t 2 cos t 3t

79. f x x3 x 1 f x is continuous on 0, 1 . f 0 1 and f 1 1 By the Intermediate Value Theorem, f x 0 for at least one value of c between 0 and 1. Using a graphing utility, we find that x 0.6823.

83. f x x2 x 1

g is continuous on 0, 1 .

f is continuous on 0, 5 .

g 0 2 > 0 and g 1 1.9 < 0.

f 0 1 and f 5 29

By the Intermediate Value Theorem, g t 0 for at least one value c between 0 and 1. Using a graphing utility, we find that t 0.5636.

1 < 11 < 29 The Intermediate Value Theorem applies. x2 x 1 11 x2 x 12 0

x 4 x 3 0 x 4 or x 3 c 3 (x 4 is not in the interval.)

Thus, f 3 11.

85. f x x3 x2 x 2

87. (a) The limit does not exist at x c.

f is continuous on 0, 3 .

(b) The function is not defined at x c.

f 0 2 and f 3 19

(c) The limit exists at x c, but it is not equal to the value of the function at x c.

2 < 4 < 19 The Intermediate Value Theorem applies.

(d) The limit does not exist at x c.

x3 x2 x 2 4 x3 x2 x 6 0

x 2 x2 x 3 0 x 2 (x2

x 3 has no real solution.) c 2

Thus, f 2 4.

89.

91. The functions agree for integer values of x:

y 5 4 3 2 1 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

g x 3 x 3 x 3 x f x 3 x 3 x x 1

3 4 5 6 7

â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;3

However, for non-integer values of x, the functions differ by 1. f x 3 x g x 1 2 x .

The function is not continuous at x 3 because lim f x 1 0 lim f x .

xâ&#x2020;&#x2019;3

for x an integer

xâ&#x2020;&#x2019;3

1 1 For example, f 2 3 0 3, g 2 3 1 4.

Section 1.4

t 2 2 t

Continuity and One-Sided Limits

N

93. N t 25 2 t

0

1

1.8

2

3

3.8

N t

50

25

5

50

25

5

Number of units

50 40 30 20 10

t 2

Discontinuous at every positive even integer.The company replenishes its inventory every two months.

4

6

8

10 12

Time (in months)

95. Let V 43 r 3 be the volume of a sphere of radius r.

V 1 43 4.19 V 5 3 53 523.6 4

Since 4.19 < 275 < 523.6, the Intermediate Value Theorem implies that there is at least one value r between 1 and 5 such that V r 275. (In fact, r 4.0341.)

97. Let c be any real number. Then lim f x does not exist since there are both rational and xâ&#x2020;&#x2019;c

irrational numbers arbitrarily close to c. Therefore, f is not continuous at c. y

1, if x < 0 0, if x 0 99. sgn x 1, if x > 0

4 3 2 1

(a) lim sgn x 1

â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;1

xâ&#x2020;&#x2019;0

x 1

2

3

4

â&#x2C6;&#x2019;2

(b) lim sgn x 1

â&#x2C6;&#x2019;3

xâ&#x2020;&#x2019;0

â&#x2C6;&#x2019;4

(c) lim sgn x does not exist. xâ&#x2020;&#x2019;0

101. True; if f x g x , x c, then lim f x lim g x and xâ&#x2020;&#x2019;c

xâ&#x2020;&#x2019;c

103. False; f 1 is not defined and lim f x does not exist. xâ&#x2020;&#x2019;1

at least one of these limits (if they exist) does not equal the corresponding function at x c.

105. (a) f x

b

0 â&#x2030;¤ x < b b < x â&#x2030;¤ 2b

0

(b) g x

y

2b

x 2

0 â&#x2030;¤ x â&#x2030;¤ b

b

x 2

b < x â&#x2030;¤ 2b

y

2b

b

x b

b

2b

NOT continuous at x b.

x b

2b

Continuous on 0, 2b .

41

42

Chapter 1

107. f x

Limits and Their Properties

x c2 c

x

, c > 0

Domain: x c2 ≥ 0 ⇒ x ≥ c2 and x 0, c2, 0 0, x c2 c

lim

x

x→0

lim

x c2 c

x→0

x

x c2 c x c2 c

x c2 c2 1 1 lim

x→0 x x c2 c x→0 x c2 c 2c

lim

Define f 0 1 2c to make f continuous at x 0.

109. h x x x

15

h has nonremovable discontinuities at x ± 1, ± 2, ± 3, . . . . −3

3 −3

Section 1.5 1.

Infinite Limits

lim 2

x

x2 4

lim 2

x

x2 4

x→ 2

x→ 2

3.

lim tan

x

4

lim tan

x

4

x→ 2

x→ 2

5. f x

1 x2 9

x

3.5

3.1

3.01

3.001

2.999

2.99

2.9

2.5

f x

0.308

1.639

16.64

166.6

166.7

16.69

1.695

0.364

lim f x

x→ 3

lim f x

x→ 3

7. f x

x2 x2 9

x

3.5

3.1

3.01

3.001

2.999

2.99

2.9

2.5

f x

3.769

15.75

150.8

1501

1499

149.3

14.25

2.273

lim f x

x→ 3

lim f x

x→ 3

Section 1.5

9. lim x→0

1 1

lim x2 x→0 x2

Therefore, x 0 is a vertical asymptote.

Infinite Limits

43

x2 2

x 2 x 1

11. lim x→2

x2 2

x 2 x 1

lim

x→2

Therefore, x 2 is a vertical asymptote. lim

x2 2

x 2 x 1

lim

x2 2

x 2 x 1

x→ 1

x→ 1

Therefore, x 1 is a vertical asymptote.

13.

lim

x→ 2

x2

x2 x2 and lim 2

x→ 2 x 4 4

15. No vertical asymptote since the denominator is never zero.

Therefore, x 2 is a vertical asymptote. lim

x→2

x2 x2 and lim 2

x→2 x 4 x2 4

Therefore, x 2 is a vertical asymptote.

17. f x tan 2x x

21.

lim

x→ 2

lim

x→ 2

sin 2x has vertical asymptotes at cos 2x

2n 1 n

, n any integer.

4 4 2 x

x 2 x 1

x

x 2 x 1

Therefore, x 2 is a vertical asymptote. lim

x

x 2 x 1

lim

x

x 2 x 1

x→1

x→1

19. lim 1 t→0

4 4 lim 1 2 t→0 t2 t

Therefore, t 0 is a vertical asymptote.

x3 1 x 1 x2 x 1

x 1 x 1

23. f x

has no vertical asymptote since lim f x lim x2 x 1 3

x→ 1

x→ 1

Therefore, x 1 is a vertical asymptote.

25. f x

x 5 x 3 x 3

,x 5 x 5 x2 1 x2 1

No vertical asymptotes. The graph has a hole at x 5.

27. s t

t has vertical asymptotes at t n , n sin t

a nonzero integer. There is no vertical asymptote at t 0 since lim t→0

t 1. sin t

44

Chapter 1

Limits and Their Properties

x2 1 lim x 1 2 x→ 1 x 1 x→ 1

29. lim

31.

lim

x2 1

x 1

lim

x2 1

x 1

x→ 1

2

−3

x→ 1

3

8

−3

3

Vertical asymptote at x 1

−8

−5

Removable discontinuity at x 1

33. lim x→2

37.

x 3

x 2

lim

x→ 3

45. lim

x→

x→3

x2 2x 3 x 1 4 lim

x→ 3 x 2 x2 x 6 5

41. lim 1 x→0

35. lim

1

x

x2 x x 1 lim 2

x→1 x 1 x 1 x→1 x 1 2

39. lim

43. lim x→0

x lim x sin x 0 csc x x→

x2

x 3 x 3

47.

2

2

sin x

lim

x→ 1 2

x sec x and

lim

x→ 1 2

x sec x .

Therefore, lim x sec x does not exist. x→ 1 2

49. f x

x2 x 1 x3 1

lim f x lim

x→1

x→1

51. f x

1

x 1

1 x2 25

lim f x

x→5

0.3

3

−8

−4

8

5

−0.3

−3

53. A limit in which f x increases or decreases without bound as x approaches c is called an infinite limit. is not a number. Rather, the symbol

55. One answer is f x

x 3 x 3

. x 6 x 2 x2 4x 12

lim f x

x→c

says how the limit fails to exist.

57.

k , 0 < r < 1. Assume k 0.

1 r k lim S lim (or if k < 0) r→1 r→1 1 r

59. S

y 3 2 1 −2

x

−1

1 −1 −2

3

Section 1.5

61. C

528x , 0 ≤ x < 100 100 x

63. (a) r

(a) C 25 $176 million

(b) r

(b) C 50 $528 million (c) C 75 $1584 million

x

x→25

f x

1

0.5

0.2

0.1

0.01

0.001

0.0001

0.1585

0.0411

0.0067

0.0017

0

0

0

0.5

lim

x→0 − 1.5

x sin x 0 x

1.5

− 0.25

(b)

x f x

1

0.5

0.2

0.1

0.01

0.001

0.0001

0.1585

0.0823

0.0333

0.0167

0.0017

0

0

0.001

0.0001

0.1667 0.1667

0.1667

0.25

− 1.5

lim

1.5

x→0

x sin x 0 x2

− 0.25

(c)

x f x

1

0.5

0.2

0.1

0.1585

0.1646

0.1663

0.1666

0.01

0.25

− 1.5

lim

1.5

x→0

x sin x 0.1167 1 6 x3

− 0.25

(d)

x f x

1

0.5

0.2

0.1

0.1585

0.3292

0.8317

1.6658

0.01

0.001

0.0001

16.67

166.7

1667.0

1.5

− 1.5

1.5

− 1.5

For n ≥ 3, lim x→0

x sin x . xn

lim

x→0

x sin x

x4

7 ft sec 12

2 15 3 ft sec 2 625 225

(c) lim

528 Thus, it is not possible. (d) lim x→100 100 x 65. (a)

2 7 625 49

2x 625 x2

Infinite Limits

45

46

Chapter 1

Limits and Their Properties (b) The direction of rotation is reversed.

67. (a) Because the circumference of the motor is half that of the saw arbor, the saw makes 1700 2 850 revolutions per minute.

(d)

(c) 2 20 cot 2 10 cot : straight sections.

The angle subtended in each circle is

2 2

2

(e)

2 .

0.3

0.6

0.9

1.2

1.5

L

306.2

217.9

195.9

189.6

188.5

450

Thus, the length of the belt around the pulleys is 20 2 10 2 30 2 .

0

Total length 60 cot 30 2

(f)

0, 2

Domain:

2

0

lim

â&#x2020;&#x2019; 2

L 60 188.5

(All the belts are around pulleys.) (g) lim L â&#x2020;&#x2019;0

71. False; let

69. False; for instance, let f x

1, f x x 3,

x2 1 or x 1

x 0 x 0.

The graph of f has a vertical asymptote at x 0, but f 0 3.

x g x 2 . x 1 73. Given lim f x and lim g x L: xâ&#x2020;&#x2019;c

xâ&#x2020;&#x2019;c

(2) Product:

If L > 0, then for L 2 > 0 there exists 1 > 0 such that g x L < L 2 whenever 0 < x c < 1. Thus, L 2 < g x < 3L 2. Since lim f x then for M > 0, there exists 2 > 0 such that f x > M 2 L whenever

x c

xâ&#x2020;&#x2019;c

< 2. Let be the smaller of 1 and 2. Then for 0 < x c < , we have f x g x > M 2 L L 2 M.

Therefore lim f x g x . The proof is similar for L < 0. xâ&#x2020;&#x2019;c

(3) Quotient: Let > 0 be given.

There exists 1 > 0 such that f x > 3L 2 whenever 0 < x c < 1 and there exists 2 > 0 such that g x L <

L 2 whenever 0 < x c < 2. This inequality gives us L 2 < g x < 3L 2. Let be the smaller of 1 and 2. Then for 0 <

x c

g x f x

<

3L 2

.

3L 2

Therefore, lim

xâ&#x2020;&#x2019;c

75. Given lim

xâ&#x2020;&#x2019;c

< , we have

g x 0. f x

1 0. f x

Suppose lim f x exists and equals L. Then, xâ&#x2020;&#x2019;c

lim 1 1 1 xâ&#x2020;&#x2019;c 0. xâ&#x2020;&#x2019;c f x lim f x L lim

xâ&#x2020;&#x2019;c

This is not possible. Thus, lim f x does not exist. xâ&#x2020;&#x2019;c