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Section 1.2 Limits

Evaluate the following limits. Use limit theorems, not ε - δ techniques. If any of them fail to exist, say so and say why.

    1. limx10x2100x10

    2. limx10x299x10

    3. limx10x2100x9

    4. limx10f(x), where f(x)=x2 for all x10, but f(10)=99.

    5. limx10x2+20x100

  1. limx4x216x+4ln|x|

  2. limxx2e4x14x

  3. limx3x67x5+x5x6+4x53

  4. limx5x77x5+12x7+6x63

  5. limx2x+3x3x3+2x1

  6. limx5x+2x3x3+x7

  7. limxax17+bxcx17dx3, a,b,c,d0

  8. limx3x+|13x|15x

  9. limxx63x6+5

  10. limuuu2+1

  11. limx1+3x2x2+x

  12. limx4x2+3x773x

  13. limxx292x1

  14. limx1+x1x21

  15. Let f(x)={x21|x1|if x1,4if x=1.

    Find limx1f(x).

  16. Let F(x)=2x23x|2x3|.

    1. Find limx1.5+F(x).

    2. Find limx1.5F(x).

    3. Does limx1.5F(x) exist? Provide a reason.

  17. limx22|x|2+x

  18. limx2|x24|105x

  19. limx4|x4|(x4)2

  20. limx8(x8)(x+2)|x8|

  21. limx2(1x2+5x+61x2)

  22. limx1x2x23x2x1

  23. limx16x4x16

  24. limx83x2x8

  25. limx42x4xx2

  26. limx01+2x14xx

  27. Find constants a and b such that limx0ax+b2x=1.

  28. limx5ex5x12

  29. limx7ex+23x7

  30. limt0sint+11t

  31. limx8x1/32x8

  32. limx(x2+xx)

  33. limx(x2+5xx2+2x)

  34. limx(x2x+1x2+1)

  35. limx(x2+3x2x)

  36. Is there a number b such that limx2bx2+15x+15+bx2+x2 exists? If so, find the value of b and the value of the limit.

  37. Determine the value of a so that f(x)=x2+ax+5x+1 has a slant asymptote y=x+3.

  38. Prove that f(x)=lnxx has a horizontal asymptote y=0.

  39. Let I be an open interval such that 4I and let a function f be defined on a set D=I{4}. Evaluate limx4f(x), where x+2f(x)x210 for all xD.

  40. Evaluate limx1f(x), where 2x1f(x)x2 for all x in the interval (0,2).

  41. Use the squeeze theorem to show that

    1. limx0x4sin(1x)=0,

    2. limx0+(xesin(1/x))=0.

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

  42. limx0+[(x2+x)1/3sin(1x2)]

  43. limx0xsin(ex)

  44. limx0xsin(1x2)

  45. limx0x2+xsin(πx)

  46. limx0xcos2(1x2)

  47. limxπ/2+xcotx

  48. limx01ex1x

  49. limx0e2x12xx2

  50. limx2exe2cos(πx2)+1

  51. limx1x21e1x71

  52. limx0ex2cos(x2)x2

  53. limx1x761x451

  54. limx1xa1xb1, a,b0

  55. limx0(sinx)100x99sin2x

  56. limx0x100sin7x(sinx)99

  57. limx0x100sin7x(sinx)101

  58. limx0arcsin3xarcsin5x

  59. limx0sin3xsin5x

  60. limx0x3sin(1x2)sinx

  61. limx0sinxxsin4x

  62. limx01cosxxsinx

  63. limθ3π2cosθ+1sinθ

  64. limxπ2(xπ2)tanx

  65. limxxtan(1/x)

  66. limx0(1sinx1x)

  67. limx0xsinxx3

  68. limx0(cscxcotx)

  69. limx0+(sinx)(lnsinx)

  70. limx(xlnx1x+1)

  71. limxex10x3

  72. limxlnxx

  73. limxln3xx2

  74. limx(lnx)2x

  75. limx1lnxx

  76. limx0ln(2+2x)ln2x

  77. limxln((2x)1/2)ln((3x)1/3)

  78. limx0ln(1+3x)2x

  79. limx1ln(1+3x)2x

  80. limθπ2+ln(sinθ)cosθ

  81. limx11x+lnx1+cos(πx)

  82. limx0(1x21tanx)

  83. limx0+(1x1ex1)

  84. limx0(coshx)1x2

  85. limx0+(cosx)1x

  86. limx0+(cosx)1x2

  87. limx0+xx

  88. limx0+xx

  89. limx0+xtanx

  90. limx0+(sinx)tanx

  91. limx0(1+sinx)1x

  92. limx(x+sinx)1x

  93. limxx1x

  94. limx(1+1x)2x

  95. limx(1+sin3x)x

  96. limx0+(x+sinx)1x

  97. limx0+(xx+1)x

  98. limxe+(lnx)1xe

  99. limxe+(lnx)1x

  100. limx0exsin(1/x)

  101. limx0(12x)1/x

  102. limx0+(1+7x)1/5x

  103. limx0+(1+3x)1/8x

  104. limx0(1+x2)3/x

  105. Let x1=100, and for n1, let xn+1=12(xn+100xn). Assume that L=limnxn exists. Calculate L.

    1. Find limx01cosxx2, or show that it does not exist.

    2. Find limx2π1cosxx2, or show that it does not exist.

    3. Find limx1arcsinx, or show that it does not exist.

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

    1. limh0416+h22h

    2. limx1lnxsin(πx)

    3. limuuu2+1

    4. limx0(12x)1/x

    5. limx0(sinx)100x99sin(2x)

    6. limx1.01xx100

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

    1. limx0(2+x)201622016x

    2. limx(x2+xx)

    3. limx0cot(3x)sin(7x)

    4. limx0+xx

    5. limxx2ex

    6. limx3sinxxx3

  108. Evaluate the following limits, if they exist.

    1. limx0f(x)|x| given that limx0xf(x)=3.

    2. limx1sin(x1)x2+x2

    3. limxx2+4x4x+1

    4. limxx4+2x44

    5. limx(ex+x)1/x

  109. Evaluate the following limits, if they exist.

    1. limx4[1x24x4]

    2. limx1x21e1x21

    3. limx0(sinx)(lnx)

  110. Evaluate the following limits. Use “” or “” where appropriate.

    1. limx1x+1x21

    2. limx0sin6x2x

    3. limx0sinh2xxex

    4. limx0+(x0.01lnx)

  111. Use the εδ definition of limits to prove that

    limx0x3=0.
    1. Sketch an approximate graph of f(x)=2x2 on [0,2]. Next, draw the points P(1,0) and Q(0,2). When using the precise definition of limx1f(x)=2, a number δ and another number ε are used. Show points on the graph which these values determine. (Recall that the interval determined by δ must not be greater than a particular interval determined by ε.)

    2. Use the graph to find a positive number δ so that whenever |x1|<δ it is always true that |2x22|<14.

    3. State exactly what has to be proved to establish this limit property of the function f.

  112. 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.

  113. If limxa[f(x)+g(x)]=2 and limxa[f(x)g(x)]=1 find limxa[f(x)g(x)].

  114. If f is continuous, use L'Hospital's rule to show that

    limh0f(x+h)f(xh)2h=f(x).

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