SECTION 14.3 One-sided Limits; Continuity 959 25. Does f x lim x 4 ( ) → exist? If it does, what is it? 26. Does f x lim x 0 ( ) → exist? If it does, what is it? 27. Is f continuous at 4? − 28. Is f continuous at 6? − 29. Is f continuous at 0? 30. Is f continuous at 2? 31. Is f continuous at 4? 32. Is f continuous at 5? In Problems 33–44, find the one-sided limit. 33. x lim 2 3 x 1 ( ) + → + 34. x lim 4 2 x 2 ( ) − → − 35. x x lim 2 5 x 1 3 ( ) + → − 36. x lim 3 8 x 2 2 ( ) − →− + 37. x lim sin x 2 π→ + 38. x lim 3cos x ( ) π→ − 39. x x lim 4 2 x 2 2 − − → + 40. x x x lim 1 x 1 3 − − → − 41. x x lim 1 1 x 1 2 3 − + →− − 42. x x x x lim x 0 3 2 4 2 − + → + 43. x x x x lim 2 2 x 2 2 2 + − + →− + 44. x x x x lim 12 4 x 4 2 2 + − + →− − In Problems 45–60, determine whether f is continuous at c. 45. f x x x x c 3 2 6 2 3 2 ( ) = − + − = 46. f x x x c 3 6 5 3 2 ( ) = − + = − 47. f x x x c 5 6 3 2 ( ) = + − = 48. f x x x c 8 4 2 3 2 ( ) = − + = 49. ( ) = + − = f x x x c 3 3 3 50. f x x x c 6 6 6 ( ) = − + = − 51. f x x x x x c 3 3 0 3 2 ( ) = + − = 52. f x x x x x c 6 6 0 2 2 ( ) = − + = 53. f x x x x x x x c 3 3 if 0 1 if 0 0 3 2 ( ) = + − ≠ = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ = 54. f x x x x x x x c 6 6 if 0 2 if 0 0 2 2 ( ) = − + ≠ − = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ = 55. f x x x x x x x c 3 3 if 0 1 if 0 0 3 2 ( ) = + − ≠ − = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ = 56. f x x x x x x x c 6 6 if 0 1 if 0 0 2 2 ( ) = − + ≠ − = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ = 57. f x x x x x x x c 1 1 if 1 2 if 1 3 1 if 1 1 3 2 ( ) = − − < = + > ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ = 58. f x x x x x x x x x c 2 2 if 2 2 if 2 4 1 if 2 2 2 ( ) = − − < = − − > = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ 59. f x e x x x x x x c 2 if 0 2 if 0 2 if 0 0 x 3 2 2 ( ) = < = + > = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ 60. f x x x x x x x x c 3cos if 0 3 if 0 3 if 0 0 3 2 2 ( ) = < = + > = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ In Problems 61–72, find the numbers at which f is continuous. At which numbers is f discontinuous? 61. f x x2 3 ( ) = + 62. f x x 4 3 ( ) = − 63. f x x x 3 2 ( ) = + 64. f x x3 7 3 ( ) = − + 65. f x x 4 sin ( ) = 66. f x x 2 cos ( ) = − 67. f x x 2 tan ( ) = 68. f x x 4 csc ( ) = 69. f x x x 2 5 4 2 ( ) = + − 70. f x x x 4 9 2 2 ( ) = − − 71. f x x x 3 ln ( ) = − 72. f x x x ln 3 ( ) = − In Problems 73–76, discuss whether R is continuous at each number c. Use limits to analyze the graph of R at c. Graph R. 73. R x x x c c 1 1 , 1 and 1 2 ( ) = − − = − = 74. R x x x c c 3 6 4 , 2 and 2 2 ( ) = + − = − = 75. R x x x x c c 1 , 1 and 1 2 2 ( ) = + − = − = 76. R x x x x c c 4 16 , 4 and 4 2 2 ( ) = + − = − =
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