2. 1 Mrs. Kessler Calculus Finding Limits Analytically 1.3 part 2 Calculus

3. 2 Basic Properties

4. 3 Limit Properties

5. 4 Thank you for not dividing by zero. What happens when you "sub in" the value of c in the and the denominator equals zero??? For example, this limit.

6. 5 New Techniques to find Limits 2. Rationalizing the numerator 1. Dividing out 3. Special cases

7. 6 Dividing Out Technique: Factor, then reduce. = Since we are taking the limit as x approaches 5, and not at x = 5, we do not have to worry about dividing by zero. Example 1:

8. 7 Dividing Out Technique: Factor, then reduce Since we are again taking the limit as x approaches 0, and not at x = 0, we do not have to worry about dividing by zero. Direct substitution yields the indeterminate form 0/0. Factor Example 2:

9. 8 Rationalizing Technique We rationalize the numerator instead of the denominator. We are still multiplying by one, thereby not changing the value, just the look. Example 3:

10. 9 What happens when you substitute x = 2? Use synthetic to simplify and divide. Example 8:

11. 10 Transcendental Limits

12. 11 Special Cases Theorem 1. 8 The Squeeze Theorem If h(x) < f(x) < g(x) for all x in an open interval containing c, except possible at c itself, and if

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

14. Example Find the limit if it exists: Substitution gives the indeterminate form…

15. Example Find the limit if it exists: Factor and cancel doesn’t work…

16. Example Find the limit if it exists: Maybe…the squeeze theorem…

17. Example g(  )=1 h(  )= cos 

18. Example & therefore…

19. Two Special Trig Limits Memorize

20. 19 The proof is in the book, and uses the squeeze theorem. You must learn these! Special limits whose proofs use the squeeze theorem

21. 20 Example 4: Rewrite = (1)(0) = 0

22. 21 Direct substitution gives 0/0 which is indeterminate. Rewrite. Example 5:

23. 22 Multiply the numerator and the denominator by 5. = 5(1) = 5 Special case Example 6:

24. 23 Rewrite Example 7:

25. 24 Sometimes you have to be creative when determining which method to use and rely upon all previous mathematics.