2. Warm-Up

3. 1-3: Evaluating Limits Analytically ©2002 Roy L. Gover (www.mrgover.com) Objectives: Find limits when substitution doesn’t work Learn about the Squeeze Theorem

4. Example Find the limit if it exists: Try substitution Substitution doesn’t work…does this mean the limit doesn’t exist? Try the factor and cancellation technique

5. Important Idea and are the same except at x =-1

6. Important Idea The functions have the same limit as x  -1

7. Procedure 1.Try substitution 2. Factor and cancel if substitution doesn’t work 3.Try substitution again The factor & cancellation technique

8. Try This Find the limit if it exists: 5 Isn’t that easy? Did you think calculus was going to be difficult?

9. Try This Find the limit if it exists:

10. Try This Find the limit if it exists: The limit doesn’t exist Confirm by graphing

11. Important Idea If substitution results in an a /0 fraction where a  0, the limit doesn’t exist.

12. Definition When substitution results in a 0/0 fraction, the result is called an indeterminate form.

13. Important Idea The limit of an indeterminate form exists, but to find it you must use a technique, such as the factor and cancel technique.

14. Try This Find the limit if it exists: -5

15. Example Find the limit if it exists: Try substitution With substitution, you get an indeterminate form Try factor & cancel Factor & cancel doesn’t work Horrible Occurrence!!! The rationalization technique to the rescue… Rationalizing the numerator allows you to factor & cancel and then substitute

16. BC warm-Up Find the limit if it exists:

17. Try This Find the limit if it exists:

18. The Squeeze Theorem Let f(x) be between g(x) & h(x) in an interval containing c. If then: f(x) is “squeezed” to L

19. Example Find the limit if it exists: Where  is in radians and in the interval

20. Example Find the limit if it exists: Substitution gives the indeterminate form… Factor and cancel or rationalization doesn’t work… Maybe…the squeeze theorem…

21. Example g(  )=1 h(  )= cos 

22. Example & therefore…

23. Two Special Trig Limits Memorize

24. Example Find the limit if it exists:

25. Example Find the limit if it exists:

26. Try This Find the limit if it exists: 0

27. Lesson Close Write, in outline form, the procedures for finding limits when substitution doesn’t work.

28. Assignment 68/45 – 61 odd