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

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

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

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 5 Quick Review

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

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 7 Quick Review Solutions

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 8 Quick Review Solutions

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 9 Quick Review Solutions

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

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

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

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 13 Example Horizontal Asymptote

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

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 15

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 16 Properties of Limits as x→±∞

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 17 Properties of Limits as x→±∞ Product Rule: Constant Multiple Rule:

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 18 Properties of Limits as x→±∞

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

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 20 Infinite Limits as x→a

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Use a graph and/or a table to determine the limit of the function below. Slide 2- 21

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

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Find the vertical asymptotes and determine the behavior of the function to the left and right of each vertical asymptote. Slide 2- 23

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Find the vertical asymptotes and determine the behavior of the function to the left and right of each vertical asymptote. Slide 2- 24

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 25 End Behavior Models

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 26 Example End Behavior Models

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 27 End Behavior Models

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 28 End Behavior Models

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine an end behavior model for the function below. Slide 2- 29

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine an end behavior model for the function below. Slide 2- 30

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine an end behavior model for the function below. Slide 2- 31

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine a power function end behavior model for the function below and determine any horizontal asymptotes. Slide 2- 32

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine a power function end behavior model for the function below and determine any horizontal asymptotes. Slide 2- 33

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Find a simple basic function as a right end behavior model and a simple basic function as a left end behavior model. Slide 2- 34

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

37. Slide 2- 37 Example “Seeing” Limits as x→±∞

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 38

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

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 Slide 2- 41 Quick Quiz Sections 2.1 and 2.2

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 42 Quick Quiz Sections 2.1 and 2.2

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 43 Quick Quiz Sections 2.1 and 2.2

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 44 Quick Quiz Sections 2.1 and 2.2

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 45 Quick Quiz Sections 2.1 and 2.2

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Assignment 2.2 pages 76 – 77, 3 – 54 multiples of 3, 59, 65, 68 and 70 Slide 2- 46