1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 2 Limits and Continuity

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 3 Example Limits

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 4 Example Limits [-6,6] by [-10,10]

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 5 One-Sided and Two-Sided Limits

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 6 One-Sided and Two-Sided Limits continued

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 7 Example One-Sided and Two-Sided Limits o 12 3 4 Find the following limits from the given graph.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 8 Sandwich Theorem

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 9 Sandwich Theorem

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.2 Limits Involving Infinity

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 11 Quick Review

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 12 Quick Review

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 13 Quick Review

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 14 Quick Review

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 15 Quick Review Solutions [-12,12] by [-8,8][-6,6] by [-4,4]

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 16 Quick Review Solutions

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 17 Quick Review Solutions

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 18 Quick Review Solutions

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 19 What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞ …and why Limits can be used to describe the behavior of functions for numbers large in absolute value.

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 20 Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line.

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 21 Horizontal Asymptote

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 22 [-6,6] by [-5,5] Example Horizontal Asymptote

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 23 Example Sandwich Theorem Revisited

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 24 Properties of Limits as x→±∞

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 25 Properties of Limits as x→±∞ Product Rule: Constant Multiple Rule:

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 26 Properties of Limits as x→±∞

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 27 Infinite Limits as x→a

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 28 Vertical Asymptote

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 29 Example Vertical Asymptote [-6,6] by [-6,6]

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 30 End Behavior Models

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 31 Example End Behavior Models

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 32 End Behavior Models

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 33 End Behavior Models

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 34 Example “Seeing” Limits as x→±∞

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 35 Quick Quiz Sections 2.1 and 2.2

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 36 Quick Quiz Sections 2.1 and 2.2

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 37 Quick Quiz Sections 2.1 and 2.2

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 38 Quick Quiz Sections 2.1 and 2.2

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 39 Quick Quiz Sections 2.1 and 2.2

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 40 Quick Quiz Sections 2.1 and 2.2

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.3 Continuity

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 42 Quick Review

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 43 Quick Review

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 44 Quick Review

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 45 Quick Review

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 46 Quick Review Solutions

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 47 Quick Review Solutions

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 48 Quick Review Solutions

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 49 Quick Review Solutions

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 50 What you’ll learn about Continuity at a Point Continuous Functions Algebraic Combinations Composites Intermediate Value Theorem for Continuous Functions …and why Continuous functions are used to describe how a body moves through space and how the speed of a chemical reaction changes with time.

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 51 Continuity at a Point

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 52 Example Continuity at a Point o

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 53 Continuity at a Point

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 54 Continuity at a Point If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f.

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 55 Continuity at a Point The typical discontinuity types are: a)Removable(2.21b and 2.21c) b)Jump(2.21d) c)Infinite(2.21e) d)Oscillating (2.21f)

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 56 Continuity at a Point

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 57 Example Continuity at a Point [-5,5] by [-5,10]

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 58 Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval.

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 59 Continuous Functions [-5,5] by [-5,10]

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 60 Properties of Continuous Functions

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 61 Composite of Continuous Functions

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 62 Intermediate Value Theorem for Continuous Functions

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 63 Intermediate Value Theorem for Continuous Functions The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches.

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.4 Rates of Change and Tangent Lines

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 65 Quick Review

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 66 Quick Review

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 67 Quick Review

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 68 Quick Review Solutions

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 69 Quick Review Solutions

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 70 Quick Review Solutions

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 71 What you’ll learn about Average Rates of Change Tangent to a Curve Slope of a Curve Normal to a Curve Speed Revisited …and why The tangent line determines the direction of a body’s motion at every point along its path.

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 72 Average Rates of Change The average rate of change of a quantity over a period of time is the amount of change divided by the time it takes. In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval. Also, the average rate of change can be thought of as the slope of a secant line to a curve.

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 73 Example Average Rates of Change

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 74 Tangent to a Curve In calculus, we often want to define the rate at which the value of a function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a. The problem with this is that we only have one point and our usual definition of slope requires two points.

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 75 Tangent to a Curve The process becomes: 1.Start with what can be calculated, namely, the slope of a secant through P and a point Q nearby on the curve. 2.Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. 3.Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 76 Example Tangent to a Curve

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 77 Example Tangent to a Curve

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 78 Slope of a Curve To find the tangent to a curve y = f(x) at a point P(a,f(a)) calculate the slope of the secant line through P and a point Q(a+h, f(a+h)). Next, investigate the limit of the slope as h→0. If the limit exists, it is the slope of the curve at P and we define the tangent at P to be the line through P with this slope.

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 79 Slope of a Curve

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 80 Slope of a Curve at a Point

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 81 Slope of a Curve

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 82 Normal to a Curve The normal line to a curve at a point is the line perpendicular to the tangent at the point. The slope of the normal line is the negative reciprocal of the slope of the tangent line.

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 83 Example Normal to a Curve

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 84 Speed Revisited

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 85 Quick Quiz Sections 2.3 and 2.4

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 86 Quick Quiz Sections 2.3 and 2.4

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 87 Quick Quiz Sections 2.3 and 2.4

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 88 Quick Quiz Sections 2.3 and 2.4

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 89 Quick Quiz Sections 2.3 and 2.4

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 90 Quick Quiz Sections 2.3 and 2.4

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 91 Chapter Test

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 92 Chapter Test

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 93 Chapter Test

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 94 Chapter Test

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 95 Chapter Test

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 96 Chapter Test

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 97 Chapter Test Solutions

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 98 Chapter Test Solutions

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 99 Chapter Test Solutions

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 100 Chapter Test Solutions

101. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 101 Chapter Test Solutions

102. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 102 Chapter Test Solutions

103. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 103 Chapter Test Solutions

104. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 104 Chapter Test Solutions