1. Calculus Finding Limits Analytically 1.3 part 2

2. Limit Properties

3. 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.

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

5. Dividing Out Technique: Factor, then reduce.
Example 1:
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. =

6. Dividing Out Technique: Factor, then reduce
Example 2:
Direct substitution yields the indeterminate form 0/0.
Factor
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.

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

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

9. Transcendental Limits

10. Special Cases
Theorem 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

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

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

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

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

15. Example
g()=1
h()=cos

16. Example
&
therefore…

17. Two Special Trig Limits
Memorize

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

19. Example 4:
Rewrite
= (1)(0) = 0

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

21. = 5(1) = 5 Multiply the numerator and the denominator by 5. Example 6:
Special case
= 5(1) = 5

22. Example 7:
Rewrite

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