1. Homework Homework Assignment #5 Read Section 2.6 Page 97, Exercises: 1 – 49 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Example, Page 97 Show that the limit leads to an indeterminate form. Then transform the function algebraically and evaluate. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Example, Page 97 25. Continued. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Example, Page 97 Use the identity: a 3 – b 3 =(a – b)(a 2 + ab + b 2 ). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Example, Page 97 Use the identity: a 3 – b 3 =(a – b)(a 2 + ab + b 2 ). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Example, Page 97 Evaluate the limit in terms of the constants involved. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

15. Example, Page 97 Evaluate the limit in terms of the constants involved. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 2: Limits Section 2.6: Trigonometric Limits Jon Rogawski Calculus, ET First Edition

17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 1 shows a “trapped” function f (x) such that l (x) ≤ f (x) ≤ u (x) for all x Figure 2 shows a “squeezed” function f (x) such that l (x) ≤ f (x) ≤ u (x) for x ≠ c and l (c) = f (c) = u (c).

18. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

19. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 3 illustrates how the function y = x sin (1/x) is squeezed as x approaches 0. Mathematically,

20. Example, Page 102 Use the Squeeze Theorem to evaluate the limit. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The above diagram leads us to the following conclusion:

22. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company From the results on the previous slide, we state Theorem 3. The graph of Theorem 3 is shown in Figure 4.

23. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The first part of Theorem 2 follows from the “squeeze” cited in Theorem 3. Part 2 results from:

24. Example, Page 102 Evaluate the limit using Theorem 2 as necessary. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

25. Example, Page 102 Evaluate the limit. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

26. Example, Page 102 Evaluate the limit. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

27. Homework Homework Assignment #6 Read Section 2.7 Page 102, Exercises: 1 – 45 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company