1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 8.1 Sequences

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 5 Quick Review

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 6 Quick Review Solutions

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 7 Quick Review Solutions

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 8 What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence …and why Sequences arise frequently in mathematics and applied fields.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 9 Defining a Sequence

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 10 Example Defining a Sequence Explicitly

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 11 Example Defining a Sequence Recursively

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 12 Arithmetic Sequence

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 13 Example Defining Arithmetic Sequences

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 14 Geometric Sequence

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 15 Example Defining Geometric Sequences

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 16 Example Constructing a Sequence

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 17 Example Graphing a Sequence Using Parametric Mode

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 18 Example Graphing a Sequence Using Sequence Graphing Mode

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 19 Example Graphing a Sequence Using Sequence Graphing Mode

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 20 Limit

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 21 Properties of Limits

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 22 Example Finding the Limit of a Sequence

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 23 The Sandwich Theorem for Sequences

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 24 Absolute Value Theorem

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 8.2 L’Hôpital’s Rule

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 26 Quick Review

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 27 Quick Review

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 28 Quick Review Solutions

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 29 Quick Review Solutions

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 30 What you’ll learn about Indeterminate Form 0/0 Indeterminate Forms ∞/∞, ∞·0, ∞-∞ Indeterminate Form 1 ∞, 0 0, ∞ 0 …and why Limits can be used to describe the behavior of functions and l’Hôpital’s Rule is an important technique for finding limits.

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 31 Indeterminate Form 0/0

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 32 L’Hôpital’s Rule (First Form)

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 33 Example Indeterminate Form 0/0

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 34 L’Hôpital’s Rule (Stronger Form)

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 35 Example Using L’Hôpital’s Rule with One-Sided Limits

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 36 Example Working with Indeterminate Form ∞/∞

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 37 Example Working with Indeterminate Form ∞·0

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 38 Indeterminate Forms 1 ∞, 0 0,∞ 0

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 39 Example Working with Indeterminate Form 1 ∞

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 40 Example Working with Indeterminate Form 1 ∞

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 41 Example Working with Indeterminate Form 0 0

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 42 Example Working with Indeterminate Form ∞ 0

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 43 Quick Quiz for Sections 8.1 and 8.2

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 44 Quick Quiz for Sections 8.1 and 8.2

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 45 Quick Quiz for Sections 8.1 and 8.2

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 46 Quick Quiz for Sections 8.1 and 8.2

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 47 Quick Quiz for Sections 8.1 and 8.2

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 48 Quick Quiz for Sections 8.1 and 8.2

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 8.3 Relative Rates of Growth

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 50 Quick Review

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 51 Quick Review

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 52 Quick Review Solutions

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 53 Quick Review Solutions

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 54 What you’ll learn about Comparing Rates of Growth Using L’Hôpital’s Rule to Compare Growth Rates Sequential versus Binary Search …and why Understanding growth rates as x→∞ is an important feature in understanding the behavior of functions.

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 55 Faster, Slower, Same-rate Growth as x→∞

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 56 Example Comparing e x and x 3 as x→∞

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 57 Example Comparing ln x with x as x→∞

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 58 Example Comparing x with x+sin x as x→∞

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 59 Transitivity of Growing Rates

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 60 Example Growing at the Same Rate as x→∞

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 61 Example Finding the Order of a Binary Search

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 8.4 Improper Integrals

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 63 Quick Review

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 64 Quick Review

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 65 Quick Review Solutions

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 66 Quick Review Solutions

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 67 What you’ll learn about Infinite Limits of Integration Integrands with Infinite Discontinuities Test for Convergence and Divergence …and why The techniques of this section allow us to extend integration techniques to cases where the interval of integration [a,b] is not finite or where integrands are not continuous.

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 68 Improper Integrals with Infinite Integration Limits

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 69 Example Evaluating an Improper Integral on [1,∞)

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 70 Example Using L’Hôpital’s Rule with Improper Integrals

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 71 Example Evaluating an Integral on (-∞,∞)

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 72 Improper Integrals with Infinite Discontinuities

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 73 Example Infinite Discontinuity at an Interior Point

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 74 Comparison Test

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 75 Example Finding the Volume of an Infinite Solid

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 76 Example Finding the Volume of an Infinite Solid

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 77 Quick Quiz Sections 8.3 and 8.4

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 78 Quick Quiz Sections 8.3 and 8.4

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 79 Quick Quiz Sections 8.3 and 8.4

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 80 Quick Quiz Sections 8.3 and 8.4

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 81 Quick Quiz Sections 8.3 and 8.4

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 82 Quick Quiz Sections 8.3 and 8.4

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 83 Chapter Test

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 84 Chapter Test

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 85 Chapter Test

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 86 Chapter Test

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 87 Chapter Test Solutions

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 88 Chapter Test Solutions

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 89 Chapter Test Solutions

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 90 Chapter Test Solution