A Collection of Problems in Differential Calculus: Problems from Calculus I Final Examinations, 2000-2020 Department of Mathematics, Simon Fraser University

Section 1.2 Limits

Evaluate the following limits. Use limit theorems, not \(\varepsilon\) - \(\delta\) techniques. If any of them fail to exist, say so and say why.

1. 1. \(\displaystyle \displaystyle \lim _{x\to 10} \frac{x^2-100}{x-10}\)

   2. \(\displaystyle \displaystyle \lim _{x\to 10} \frac{x^2-99}{x-10}\)

   3. \(\displaystyle \displaystyle \lim _{x\to 10} \frac{x^2-100}{x-9}\)

   4. \(\ds \lim _{x\to 10} f(x)\text{,}\) where \(f(x)=x^2\) for all \(x\not= 10\text{,}\) but \(f(10)=99\text{.}\)

   5. \(\displaystyle \displaystyle \lim _{x\to 10}\sqrt{-x^2+20x-100}\)

2. \(\displaystyle \displaystyle \lim _{x\to -4} \frac{x^2-16}{x+4}\ln |x|\)

3. \(\displaystyle \displaystyle \lim _{x\to \infty} \frac{x^2}{e^{4x}-1-4x}\)

4. \(\displaystyle \displaystyle \lim _{x\to -\infty} \frac{3x^6-7x^5+x}{5x^6+4x^5-3}\)

5. \(\displaystyle \displaystyle \lim _{x\to -\infty} \frac{5x^7-7x^5+1}{2x^7+6x^6-3}\)

6. \(\displaystyle \displaystyle \lim _{x\to -\infty} \frac{2x+3x^3}{x^3+2x-1}\)

7. \(\displaystyle \displaystyle \lim _{x\to -\infty} \frac{5x+2x^3}{x^3+x-7}\)

8. \(\displaystyle \lim _{x\to \infty} \frac{ax^{17}+bx}{cx^{17}-dx^3}\text{,}\) \(a,b,c,d\not=0\)

9. \(\displaystyle \displaystyle \lim _{x\to \infty} \frac{3x+|1-3x|}{1-5x}\)

10. \(\displaystyle \displaystyle \lim _{x\to -\infty} \frac{\sqrt{x^6-3}}{\sqrt{x^6+5}}\)

11. \(\displaystyle \displaystyle \lim _{u\to \infty} \frac{u}{\sqrt{u^2+1}}\)

12. \(\displaystyle \displaystyle \lim _{x\to \infty} \frac{1+3x}{\sqrt{2x^2+x}}\)

13. \(\displaystyle \displaystyle \lim _{x\to \infty} \frac{\sqrt{4x^2+3x}-7}{7-3x}\)

14. \(\displaystyle \displaystyle \lim _{x\to -\infty} \frac{\sqrt{x^2-9}}{2x-1}\)

15. \(\displaystyle \displaystyle \lim _{x\to 1^+} \frac{\sqrt{x-1}}{x^2-1}\)

16. Let \(\ds f(x)=\left\{ \begin{array}{lll} \frac{x^2-1}{|x-1|}\amp \mbox{if } \amp x\not= 1,\\ 4\amp \mbox{if } \amp x= 1. \end{array} \right.\)

    Find \(\displaystyle \lim _{x\to 1^-}f(x)\text{.}\)

17. Let \(F(x)=\frac{2x^2-3x}{|2x-3|}\text{.}\)

    1. Find \(\displaystyle \lim _{x\to 1.5^+}F(x)\text{.}\)

    2. Find \(\displaystyle \lim _{x\to 1.5^-}F(x)\text{.}\)

    3. Does \(\displaystyle \lim _{x\to 1.5}F(x)\) exist? Provide a reason.

18. \(\displaystyle \displaystyle \lim _{x\to -2} \frac{2-|x|}{2+x}\)

19. \(\displaystyle \displaystyle \lim _{x\to 2^-} \frac{|x^2-4|}{10-5x}\)

20. \(\displaystyle \displaystyle \lim _{x\to 4^-} \frac{|x-4|}{(x-4)^2}\)

21. \(\displaystyle \displaystyle \lim _{x\to 8} \frac{(x-8)(x+2)}{|x-8|}\)

22. \(\displaystyle \displaystyle \lim _{x\to 2} \left(\frac{1}{x^2+5x+6}-\frac{1}{x-2}\right)\)

23. \(\displaystyle \displaystyle \lim _{x\to -1} \frac{x^2-x-2}{3x^2-x-1}\)

24. \(\displaystyle \displaystyle \lim _{x\to 16}\frac{\sqrt{x}-4}{x-16}\)

25. \(\displaystyle \displaystyle \lim _{x\to 8}\frac{\sqrt[3]{x}-2}{x-8}\)

26. \(\displaystyle \displaystyle \lim _{x\to 4} \frac{2-\sqrt{x}}{4x-x^2}\)

27. \(\displaystyle \displaystyle \lim _{x\to 0} \frac{\sqrt{1+2x}-\sqrt{1-4x}}{x}\)

28. Find constants \(a\) and \(b\) such that \(\displaystyle \lim _{x\to 0}\frac{\sqrt{ax+b}-2}{x}=1\text{.}\)

29. \(\displaystyle \displaystyle \lim _{x\to 5}e^{ \frac{x-5}{\sqrt{x-1}-2}}\)

30. \(\displaystyle \displaystyle \lim _{x\to 7}e^{ \frac{\sqrt{x+2}-3}{x-7}}\)

31. \(\displaystyle \displaystyle \lim _{t\to 0} \frac{\sqrt{\sin t +1}-1}{t}\)

32. \(\displaystyle \displaystyle \lim _{x\to 8}\frac{x^{1/3}-2}{x-8}\)

33. \(\displaystyle \displaystyle \lim _{x\to \infty }\left( \sqrt{x^2+x}-x\right)\)

34. \(\displaystyle \displaystyle \lim _{x\to -\infty }\left( \sqrt{x^2+5x}-\sqrt{x^2+2x}\right)\)

35. \(\displaystyle \displaystyle \lim _{x\to \infty }\left( \sqrt{x^2-x+1}-\sqrt{x^2+1}\right)\)

36. \(\displaystyle \displaystyle \lim _{x\to \infty }\left( \sqrt{x^2+3x-2}-x\right)\)

37. Is there a number \(b\) such that \(\ds \lim _{x\to -2}\frac{bx^2+15x+15+b}{x^2+x-2}\) exists? If so, find the value of \(b\) and the value of the limit.

38. Determine the value of \(a\) so that \(\ds f(x)=\frac{x^2+ax+5}{x+1}\) has a slant asymptote \(y=x+3\text{.}\)

39. Prove that \(f(x)=\frac{\ln x}{x}\) has a horizontal asymptote \(y=0\text{.}\)

40. Let \(I\) be an open interval such that \(4\in I\) and let a function \(f\) be defined on a set \(D=I\backslash \{ 4\}\text{.}\) Evaluate \(\displaystyle \lim _{x\to 4}f(x)\text{,}\) where \(x+2\leq f(x)\leq x^2-10\) for all \(x\in D\text{.}\)

41. Evaluate \(\displaystyle \lim _{x\to 1}f(x)\text{,}\) where \(2x-1\leq f(x)\leq x^2\) for all \(x\) in the interval \((0,2)\text{.}\)

42. Use the squeeze theorem to show that

    1. \(\displaystyle \lim _{x\to 0}x^4\sin\left( \frac{1}{x}\right) =0\text{,}\)

    2. \(\displaystyle \lim _{x\to 0^+}\left( \sqrt{x}e^{\sin (1/x)}\right) =0\text{.}\)

    Evaluate the following limits. If any of them fail to exist, say so and say why.

43. \(\displaystyle \displaystyle \lim _{x\to 0^+}\left[ (x^2+x)^{1/3}\sin \left( \frac{1}{x^2}\right)\right]\)

44. \(\displaystyle \displaystyle \lim _{x\to 0}x\sin \left( \frac{e}{x}\right)\)

45. \(\displaystyle \displaystyle \lim _{x\to 0}x\sin \left( \frac{1}{x^2}\right)\)

46. \(\displaystyle \displaystyle \lim _{x\to 0} \sqrt{x^2+x}\cdot \sin\left(\frac{\pi}{x}\right)\)

47. \(\displaystyle \displaystyle \lim _{x\to 0} x\cos^2\left(\frac{1}{x^2}\right)\)

48. \(\displaystyle \displaystyle \lim _{x\to \pi /2^+}\frac{x}{\cot x}\)

49. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{1-e^{-x}}{1-x}\)

50. \(\displaystyle \displaystyle \lim _{x\to 0} \frac{e^{2x}-1-2x}{x^2}\)

51. \(\displaystyle \displaystyle \lim _{x\to 2} \frac{e^x-e^2}{\cos\left(\frac{\pi x}{2}\right)+1}\)

52. \(\displaystyle \displaystyle \lim _{x\to 1} \frac{x^2-1}{e^{1-x^7}-1}\)

53. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{e^{-x^2}\cos (x^2)}{x^2}\)

54. \(\displaystyle \displaystyle \lim _{x\to 1}\frac{x^{76}-1}{x^{45}-1}\)

55. \(\displaystyle \lim _{x\to 1} \frac{x^a-1}{x^b-1}\text{,}\) \(a,b\not=0\)

56. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{(\sin x)^{100}}{x^{99}\sin 2x}\)

57. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{x^{100}\sin 7x}{(\sin x)^{99}}\)

58. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{x^{100}\sin 7x}{(\sin x)^{101}}\)

59. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{\arcsin 3x}{\arcsin 5x}\)

60. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{\sin 3x}{\sin 5x}\)

61. \(\displaystyle \displaystyle \lim _{x\to 0} \frac{x^3\sin \left( \frac{1}{x^2}\right)}{\sin x}\)

62. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{\sin x}{\sqrt{x\sin 4x}}\)

63. \(\displaystyle \displaystyle \lim _{x \to 0}\frac{1-\cos x}{x\sin x}\)

64. \(\displaystyle \displaystyle \lim _{\theta\to \frac{3\pi}{2}} \frac{\cos \theta +1}{\sin\theta}\)

65. \(\displaystyle \displaystyle \lim _{x\to \frac{\pi}{2}} \left(x-\frac{\pi}{2}\right)\tan x\)

66. \(\displaystyle \displaystyle \lim _{x\to \infty }x\tan (1/x)\)

67. \(\displaystyle \displaystyle \lim _{x\to 0}\left( \frac{1}{\sin x}-\frac{1}{x}\right)\)

68. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{x- \sin x}{x^3}\)

69. \(\displaystyle \displaystyle \lim _{x\to 0} (\csc x-\cot x)\)

70. \(\displaystyle \displaystyle \lim _{x\to 0^+}(\sin x)(\ln \sin x)\)

71. \(\displaystyle \displaystyle \lim _{x\to \infty} \left(x\cdot \ln\frac{x-1}{x+1}\right)\)

72. \(\displaystyle \displaystyle \lim _{x\to \infty} \frac{e^{\frac{x}{10}}}{x^3}\)

73. \(\displaystyle \displaystyle \lim _{x\to \infty }\frac{\ln x}{\sqrt{x}}\)

74. \(\displaystyle \displaystyle \lim _{x\to \infty }\frac{\ln 3x}{x^2}\)

75. \(\displaystyle \displaystyle \lim _{x\to \infty }\frac{(\ln x)^2}{x}\)

76. \(\displaystyle \displaystyle \lim _{x\to 1 }\frac{\ln x}{x}\)

77. \(\displaystyle \displaystyle \lim _{x\to 0 }\frac{\ln (2+2x)-\ln 2}{x}\)

78. \(\displaystyle \displaystyle \lim _{x\to \infty }\frac{\ln ((2x)^{1/2})}{\ln ((3x)^{1/3})}\)

79. \(\displaystyle \displaystyle \lim _{x\to 0 }\frac{\ln (1+3x)}{2x}\)

80. \(\displaystyle \displaystyle \lim _{x\to 1 }\frac{\ln (1+3x)}{2x}\)

81. \(\displaystyle \displaystyle \lim _{\theta \to \frac{\pi }{2} ^+}\frac{\ln (\sin \theta)}{\cos \theta }\)

82. \(\displaystyle \displaystyle \lim _{x\to 1 }\frac{1-x+\ln x}{1+\cos (\pi x)}\)

83. \(\displaystyle \displaystyle \lim _{x\to 0 }\left( \frac{1}{x^2}-\frac{1}{\tan x}\right)\)

84. \(\displaystyle \displaystyle \lim _{x\to 0^+} \left(\frac{1}{x}-\frac{1}{e^x-1}\right)\)

85. \(\displaystyle \displaystyle\lim _{x\to 0}(\cosh x)^{\frac{1}{x^2}}\)

86. \(\displaystyle \displaystyle \lim _{x\to 0^+}(\cos x)^{\frac{1}{x}}\)

87. \(\displaystyle \displaystyle \lim _{x\to 0^+}(\cos x)^{\frac{1}{x^2}}\)

88. \(\displaystyle \displaystyle \lim _{x\to 0^+}x^{x}\)

89. \(\displaystyle \displaystyle \lim _{x\to 0^+}x^{\sqrt{x}}\)

90. \(\displaystyle \displaystyle \lim _{x\to 0^+} x^{\tan x}\)

91. \(\displaystyle \displaystyle \lim _{x\to 0^+}(\sin x)^{\tan x}\)

92. \(\displaystyle \displaystyle \lim _{x\to 0}(1+\sin x)^{\frac{1}{x}}\)

93. \(\displaystyle \displaystyle \lim _{x\to \infty }(x+\sin x)^{\frac{1}{x}}\)

94. \(\displaystyle \displaystyle \lim _{x\to \infty }x^{\frac{1}{x}}\)

95. \(\displaystyle \displaystyle \lim _{x\to \infty }\left( 1+ \frac{1}{x}\right) ^{2x}\)

96. \(\displaystyle \displaystyle \lim _{x\to \infty }\left( 1+\sin \frac{3}{x}\right) ^x\)

97. \(\displaystyle \displaystyle \lim _{x\to 0^+}(x+\sin x)^{\frac{1}{x}}\)

98. \(\displaystyle \displaystyle \lim _{x\to 0^+}\left( \frac{x}{x+1}\right) ^{x}\)

99. \(\displaystyle \displaystyle \lim _{x\to e^+}(\ln x)^{\frac{1}{x-e}}\)

100. \(\displaystyle \displaystyle \lim _{x\to e^+}(\ln x)^{\frac{1}{x}}\)

101. \(\displaystyle \displaystyle \lim _{x\to 0}e^{x\sin (1/x)}\)

102. \(\displaystyle \displaystyle \lim _{x\to 0}(1-2x)^{1/x}\)

103. \(\displaystyle \displaystyle \lim _{x\to 0^+}(1+7x)^{1/5x}\)

104. \(\displaystyle \displaystyle \lim _{x\to 0^+}(1+3x)^{1/8x}\)

105. \(\displaystyle \displaystyle \lim _{x\to 0}\left( 1+\frac{x}{2}\right) ^{3/x}\)

106. Let \(x_1=100\text{,}\) and for \(n\geq 1\text{,}\) let \(\displaystyle x_{n+1}=\frac{1}{2}\left(x_n+\frac{100}{x_n}\right)\text{.}\) Assume that \(\displaystyle L=\lim _{n\to \infty }x_n\) exists. Calculate \(L\text{.}\)

107. 1. Find \(\displaystyle \lim _{x \to 0}\frac{1-\cos x}{x^2}\text{,}\) or show that it does not exist.

     2. Find \(\displaystyle\lim _{x \to 2\pi }\frac{1-\cos x}{x^2}\text{,}\) or show that it does not exist.

     3. Find \(\displaystyle \lim _{x \to -1}\arcsin x\text{,}\) or show that it does not exist.

108. Compute the following limits or state why they do not exist:

     1. \(\displaystyle \displaystyle \lim _{h\to 0}\frac{\sqrt[4]{16+h}-2}{2h}\)

     2. \(\displaystyle \displaystyle \lim _{x\to 1}\frac{\ln x}{\sin (\pi x)}\)

     3. \(\displaystyle \displaystyle \lim _{u\to \infty }\frac{u}{\sqrt{u^2+1}}\)

     4. \(\displaystyle \displaystyle \lim _{x\to 0 }(1-2x)^{1/x}\)

     5. \(\displaystyle \displaystyle \lim _{x\to 0 }\frac{(\sin x)^{100}}{x^{99}\sin (2x)}\)

     6. \(\displaystyle \displaystyle \lim _{x\to \infty }\frac{1.01^x}{x^{100}}\)

109. Find the following limits. If a limit does not exist, write 'DNE'. No justification necessary.

     1. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{(2+x)^{2016}-2^{2016}}{x}\)

     2. \(\displaystyle \displaystyle \lim _{x\to \infty }(\sqrt{x^2+x}-x)\)

     3. \(\displaystyle \displaystyle \lim _{x\to 0} \cot (3x)\sin (7x)\)

     4. \(\displaystyle \displaystyle \lim _{x\to 0^+}x^x\)

     5. \(\displaystyle \displaystyle \lim _{x\to \infty} \frac{x^2}{e^x}\)

     6. \(\displaystyle \displaystyle \lim _{x\to 3}\frac{\sin x-x}{x^3}\)

110. Evaluate the following limits, if they exist.

     1. \(\displaystyle \lim _{x\to 0}\frac{f(x)}{|x|}\) given that \(\displaystyle \lim _{x\to 0}xf(x)=3\text{.}\)

     2. \(\displaystyle \displaystyle \lim _{x\to 1} \frac{\sin (x-1)}{x^2+x-2}\)

     3. \(\displaystyle \displaystyle \lim _{x\to -\infty }\frac{\sqrt{x^2+4x}}{4x+1}\)

     4. \(\displaystyle \displaystyle \lim _{x\to \infty }\frac{\sqrt{x^4+2}}{x^4-4}\)

     5. \(\displaystyle \displaystyle \lim _{x\to \infty} (e^x+x)^{1/x}\)

111. Evaluate the following limits, if they exist.

     1. \(\displaystyle \displaystyle \lim _{x\to 4}\left[ \frac{1}{\sqrt{x}-2}-\frac{4}{x-4}\right]\)

     2. \(\displaystyle \displaystyle \lim _{x\to 1} \frac{x^2-1}{e^{1-x^2}-1}\)

     3. \(\displaystyle \displaystyle \lim _{x\to 0}(\sin x)(\ln x)\)

112. Evaluate the following limits. Use “\(\infty\)” or “\(-\infty\)” where appropriate.

     1. \(\displaystyle \displaystyle \lim _{x\to 1^-}\frac{x+1}{x^2-1}\)

     2. \(\displaystyle \displaystyle \lim _{x\to 0} \frac{\sin 6x}{2x}\)

     3. \(\displaystyle \displaystyle \lim _{x\to 0}\frac{\sinh 2x}{xe^x}\)

     4. \(\displaystyle \displaystyle \lim _{x\to 0^+}(x^{0.01}\ln x)\)

113. Use the \(\varepsilon -- \delta\) definition of limits to prove that

     \begin{equation*} \lim _{x\to 0}x^3=0\text{.} \end{equation*}

114. 1. Sketch an approximate graph of \(f(x)=2x^2\) on \([0,2]\text{.}\) Next, draw the points \(P(1,0)\) and \(Q(0,2)\text{.}\) When using the precise definition of \(\lim _{x\to 1}f(x)=2\text{,}\) a number \(\delta\) and another number \(\varepsilon\) are used. Show points on the graph which these values determine. (Recall that the interval determined by \(\delta\) must not be greater than a particular interval determined by \(\varepsilon\text{.}\))

     2. Use the graph to find a positive number \(\delta\) so that whenever \(|x-1|\lt \delta\) it is always true that \(|2x^2-2|\lt \frac{1}{4}\text{.}\)

     3. State exactly what has to be proved to establish this limit property of the function \(f\text{.}\)

115. Give an example of a function \(F=f+g\) such that the limits of \(f\) and \(g\) at \(a\) do not exist and that the limit of \(F\) at \(a\) exists.

116. If \(\ds \lim_{x\to a}[f(x)+g(x)]=2\) and \(\ds \lim_{x\to a}[f(x)-g(x)]=1\) find \(\ds \lim_{x\to a}[f(x)\cdot g(x)]\text{.}\)

117. If \(f'\) is continuous, use L'Hospital's rule to show that

     \begin{equation*} \displaystyle \lim _{h\to 0}\frac{f(x+h)-f(x-h)}{2h}=f'(x)\text{.} \end{equation*}

     Explain the meaning of this equation with the aid of a diagram.