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Contents

2 Limits and Continuity 35

2.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 Numerical Introduction to Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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2 CONTENTS

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Chapter 2

Limits and Continuity

2.1 Limits

1. Define f : R Rby the rule f(x) = 101 for every x in the domain off. Find the following limits ifthey exist:

(a) limx101

f(x) (b) limx100

f(x)

(c) limx20

f(x)101f(x) (d) limx1

[f(x)]2 2f(x)

2. Letg : R Rbe defined by the ruleg(x) =x2 4 for all x in the domain ofg andL : R Rbe definedby the rule L(x) = 2x

1 for all x in the domain ofL. Find the following limits:

(a) limx2

g(x) (b) limx2

g(x)

(c) limx3

g(x) L(x) (d) limx1

[g(x) L(x)]

(e) limx1/2

g(x)L(x)

(f) limx1/2

g(x)L(x)

(g) limx2

[g(x) g(2)] (h) limx1

[g(x) g(L(0))]

3. Evaluate the following limits:

(a) limy

3|y| (b) lim

w

2|w|

(c) limt3

|t 2| (d) limy61

|y||y|

(e) limx0

(|x| | 3|) (f) limxa(|x| |a|)

(g) limz2

(|z|2 + 2|z| 3) (h) limx2

x|x|

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36 Limits and Continuity

4. Evaluate the following limits:

(a) limu 3

2

u24u2 (b) limu2

u24u2

(c) limv0

v2+v2v+2 (d) limv2

v2+v2v+2

(e) limx

4

sin2 xcos2(x)sinx+cos x (f) limx17

(sin2 x+ cos2 x)

(g) limx

2

sinx2+cos x2

sinx2

5. Find the following limits if they exist:

(a) limx2

x2x2 (b) limx3

x29x3

(c) limxa

1/x1/ax1

, a

= 0 (d) lim

x4

|x|4x4

(e) limx1

2x33x2+2x1x1 (f) limx3

x3+27x+3

6. Find the following limits if they exist:

(a) limxa

x4a4xa (b) limx0

x2xx

(c) limh0

4(x+h)34x3h (d) limx0

122x12x

(e) limx0

122x1+2x

(f) limx10

(1 log10x)

7. Find limxa

f(x)f(a)xa and limt0

f(a+t)f(a)t for:

(a) f(x) =x2, a= 3 (b) f(x) = x2, a= 1/3(b) f(x) =x2 + 1, a= 2 (d) f(x) = 3x2 x, a= 0(e) f(x) = 1/2x2 3x+ 1, a= 0 (f) f(x) = (x 3)2 5, a= 1(g) f(x) = |x|2, a= 2 (h) f(x) = x|x|, a= 2

In problems 8 to 13 find limxa

g(x)g(a)xa for the given function g and the valuea.

8. g(x) = x+ 2, a= 93 9. g(x) =

2x, a= 14

10. g(x) = x(1 x), a= 1 11. g(x) = (x+ 1)2, a= 1

12. g(x) = 2x2 + 1, a= 2 13. g(x) = 1x+1 , a= 1

Find the following limits if they exist:

14. limx1

+ |x1|x1 and limx1|x1|x1

15. limx2+

x+2|x+2| and limx2

x+2|x+2|

16. limx0+ f(x), where f(x) = x2, x

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2.1 Limits 37

17. limx0

f(x), where f(x) is defined in question 16.

18. limx1+

g(x), whereg (x) = (x 1)2, x >1

(x 1)3

, x

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26. Iff(x) = x2+x6x1 find the following limits if they exist.

(a) limx0

f(x) (b) limx1

f(x) (c) limx3

f(x)

(d) limx2 f(x) (e) limx1+ f(x) (f) limx1f(x)

(g) limx

f(x) (h) limx

f(x) (i) limx

|f(x) x 2|

Find the following limits if they exist.

27. limx+

5x1x+4 28. limx+

x2+3x+73x22x1

29. limx+

2x+5x2+1

30. limx+

2x2+4x1x3+4

31. limx

+

x2+6x+6

32. limx

+

x26x+6

33. limx+

2x23x+5

34. limx+

2x23x+4x+4

35. limx2

xx2 36. limx3

x2

9x

37. limx1+

x+1x21 38. limx1

x+1x21

39. limx3

9x2x3 40. limx0

1x 1x2

41. limx3 1x3 2x29

In problems 42 to 47 sketch the graph of the given function. Find the left and right hand limits at eachof the given values ofx, if they exist. Find the limits at the given values ofx if they exist. Does the limitequal the value of the function at the number in question? Is the function continuous at the given values ofx?

42. (a) f(x) =

x+ 6 if x 3,x ifx >3;

atx = 3; x = 1.

(b) f(x) =

3x+ 4 if x < 1,

x ifx 1; atx = 1; x = 0.

43. (a) f(x) =

x2

ifx

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2.1 Limits 39

45. (a) f(x) =

x+ 1 if x 1; atx = 1, 0.

(b) f(x) = x ifx 0,x ifx 2; atx = 2.

(b) f(x) =

x/3 ifx 10;at x = 1, 10.

In problems 49 to 62, sketch the graphs of each of the functions defined and determine if each is continuousat the given valuec. (Hint: Use definition of continuity of a function at a point.)

49. f(x) = 3x2 , at c = 2.

50. f(x) =

1

x+ 4, x = 4

0, x= 4at c = 4.

51. f(x) =

x2 3x+ 2x 1 , x = 1

5, x >1at c = 1.

52. f(x) =

3x 1, at c = 1/3.

53. f(x) =

2x+ 1, x 1x+ 4, x >1 at c = 1.

54. f(x) =

2x+ 1, x 2x+ 4, x >2

at c = 2.

55. f(x) = |x 2|, at c = 2,

56. f(x) =|

x

2

|, x

= 2

1, x= 2 at c = 2.

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2.2 Numerical Introduction to Limits 41

73. limx5

sin2(x) 74. limx0

cossinxx

75. limx

log10 x+1x 76. limxa log2 x2a2x+a

77. limx

log2

1 + 2x3

78. lim

x

log2(8x 1) + log2

1x4

79. limx0

log10sinxx

80. lim

x4

log10(tan x)

81. limx1

log10(x

2 1) log10(x 1)

82. limx0

log10(10x)

83. limx1

log2(2x+1) 84. lim

x2log3(3

x34)

85. (a) limx

2x (b) limx

2x

(c) limx0

log x (d) limx log x

86. (a) limx 2

sinx (b) limx 2

tanx

(c) limx2

1x (d) lim

x2 1

x2

87. (a) limx0

(1 2x) (b) limx0

log2(1 + 2x)

(c) limx

(1 2x) (d) limx

log2(1 2x)

2.2 Numerical Introduction to Limits

In this section we will look at the behaviour of a function near but not ata particular number which mayor may not be in its domain. By observing the values off(x) for specific values ofx near to the point inquestion we can usually get a good idea of what the limit off at that point may be. But no amount ofnumerical data can provide a proof that the limit is a specific number.

Example: Study the behaviour ofg (x) = (x3 2x2)/(x 2) near x = 2.

Solution: g(x) =x3 2x2

x 2 = x2(x 2)

(x 2) .

Observe that the only difference between g and x2 is that g(x) is not defined at x = 2 (g(2) = 00 is not areal number!). 2 is not in the domain ofg

x g(x) x g(x)1.9 3.61 2.1 4.411.99 3.9601 2.01 4.0411.999 3.996001 2.001 4.004001

Conclusion: From the given data we guess the limit is 4 even thoughg (x) is not defined at x = 2.

For the given function, f, and number b in problems 88 to 108 answer the following questions.

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(a) Choose a sequence of values ofx nearb that approachb from the left. Use these values ofx to computea table of values for f(x). Next choose a sequence of values ofx that approach b from the right andcompute another table of values off(x). On the basis of the information given in both tables guesswhich of the following statements are true:

(i) the function has a limit.

(ii) the function has a right hand limit.

(iii) the function has a left hand limit.

(iv) the function has no limit.

Guess the values of the limits you think exist.

(b) If the limxb

f(x) exists prove that it does.

88. f(x) = 1

1x; b= 1.

89. f(x) = 1 |x|x ; b= 0.

90. f(x) = 11|x| ; b= 1.

91. f(x) = 3x + 1; b= 3.

92. f(x) = 4x2

2x; b= 2.

93. f(x) = x+x6x+3 ; b= 3.

94. f(x) = x3+2x2

x2

x+2 ; b= 2.95. f(x) = x

316xx2+4x

; b= 0.

96. f(x) = x9x3 ; b= 9.

97. f(x) = cosx1x ; b= 0.

98. f(x) = (1 +x)1x ; b= 0.

99. f(x) = 21

x2 + 1; b= 0.

100. f(x) = sin 2x ;b = 0. Those people whose last names begin with AM usex = 1/2, 1/4, 1/6, . . .

to compute a table offvalues. All others use x = 1/3, 1/5, 1/7, 1/9 . . ..

101. f(x) =cosx1x2 ; b= 0.

102. f(x) =

cos x 1 x22 x4

12

/x6; b= 0.

103. f(x) = x+1sin(x+1)(x+1)3

; b= 1.

104. f(x) = (sin x x x3/6)/x5; b= 0.

105. f(x) = tanxxtanx+x ; b= 0.

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2.2 Numerical Introduction to Limits 43

106. f(x) =22 cos x|x| ; b= 0.

107. Let h andg be two functions and b

R.

Evaluate: limxb

h(x) h(b) (x b)h(b)g(x) g(b) (x b)g(b)

g(b)h(b)

.

108. f(x) = (1 + sin x)1x ; b= 0.

109. f(x) = (cos x sin x) 1x ; b= 0.

110. f(x) = arctan 1x ; b= 0.

111. f(x) = cos sinxx; b= 0.

112. f(x) = sin 3x2x

; b= 0.

113. f(x) = x3+8x+2

; b= 2.

114. f(x) = 2x1x ; b= 0.

115. f(x) = sinxsinx ; b= .

116. f(x) = 2(2x11)x1 ; b= 1.

117. f(x) = 1x

x(1x) ; b= 1.

118. f(x) = xsinxxtanx ; b= 0.

119. f(x) = tanxx ; b= 0.

120. f(x) = ex1x

cos2x1 ; b= 0.

121. f(x) = lnxsin(x1)(x1)2 ; b= 1.

122. f(x) = ln(sinx)(2x)2 ; b=

2

.

123. f(x) = ln(x/7)7x ; b= 7.

124. f(x) = xlnx 1x lnx ; b= 1.

125. f(x) = x1lnx

; b= 1.

126. f(x) = x1x+245 ; b= 1.

127. f(x) = x2+|x2|4|x2| ; b= 2.

128. f(x) = 1+x1x

1x

; b= 0.

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