1. 2-5: Techniques for Evaluating Limits
Objectives:
Find limits using direct substitution
Find limits when substitution doesn’t work
©2002 Roy L. Gover (

2. Basic Limits (on handout)

3. Properties of Limits (on handout)

4. Properties of Limits (on handout)
(n is even)
(n is odd)

5. Important Idea
The limit, if it exists, of f(x) as xc is f(c) if f(x) is continuous at c.
←use substitution

6. f(x) continuous at x=2 f(2) exists and
Example
f(2)=4
f(x) continuous at x=2 f(2) exists and
The limit is found by substitution

7. Example
Find the limit, if it exists:
= 2(27) + 1 = 55

8. Try This
Find the limit, if it exists: 3

9. Try This
Find the limit, if it exists: -2

10. Important Idea
The limit, if it exists, of f(x) as xc is not f(c) if f(x) is discontinuous at c.
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑
Cannot use substitution!!
Must be other methods 

11. There is no hope… Important Idea Horrible Occurrence!!!
If substitution results in an a/0 fraction where a0, the limit doesn’t exist.
There is no hope…
Horrible Occurrence!!!

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

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

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

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

16. Procedure Try substitution
Factor and cancel if substitution doesn’t work
Try substitution again
The factor & cancellation technique

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

18. Try This
Find the limit if it exists:

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

20. 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.

21. Horizontal Asymptotes!!
Limits as x→∞, x →-∞
Follow the rules for
Horizontal Asymptotes!!

22. Example
Find the limit, if it exists:

23. Example
Find the limit, if it exists:

24. Example
Find the limit, if it exists:

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

26. Try This
Find the limit if it exists: