1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.1 Rates of Change and Limits

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 2 What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem …and why Limits can be used to describe continuity, the derivative and the integral: the ideas giving the foundation of calculus.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 3 Average and Instantaneous Speed

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 4 Definition of Limit

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 5 Definition of Limit continued

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 6 Definition of Limit continued

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 7 Properties of Limits

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 8 Properties of Limits continued Product Rule: Constant Multiple Rule:

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 9 Properties of Limits continued

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 10 Example Properties of Limits

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 11 Polynomial and Rational Functions

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

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 13 Evaluating Limits As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions.

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

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 15 Example Limits [-6,6] by [-10,10]

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 16 One-Sided and Two-Sided Limits

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 17 One-Sided and Two-Sided Limits continued

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 18 Example One-Sided and Two-Sided Limits o 12 3 4 Find the following limits from the given graph.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 19 Sandwich Theorem

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 20 Sandwich Theorem

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

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 22 Quick Review

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 23 Quick Review

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 24 Quick Review

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 25 Quick Review

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 26 Quick Review Solutions [-12,12] by [-8,8][-6,6] by [-4,4]

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 27 Quick Review Solutions

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 28 Quick Review Solutions

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 29 Quick Review Solutions

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

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

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 32 Horizontal Asymptote

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 33 [-6,6] by [-5,5] Example Horizontal Asymptote

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 34 Example Sandwich Theorem Revisited

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 35 Properties of Limits as x→±∞

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 36 Properties of Limits as x→±∞ Product Rule: Constant Multiple Rule:

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 37 Properties of Limits as x→±∞

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 38 Infinite Limits as x→a

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 39 Vertical Asymptote

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 40 Example Vertical Asymptote [-6,6] by [-6,6]

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 41 End Behavior Models

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 42 Example End Behavior Models

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 43 End Behavior Models

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 44 End Behavior Models

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 45 Example “Seeing” Limits as x→±∞

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 46 Quick Quiz Sections 2.1 and 2.2

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 47 Quick Quiz Sections 2.1 and 2.2

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 48 Quick Quiz Sections 2.1 and 2.2

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 49 Quick Quiz Sections 2.1 and 2.2

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 50 Quick Quiz Sections 2.1 and 2.2

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 51 Quick Quiz Sections 2.1 and 2.2