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2-5: Techniques for Evaluating Limits

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Presentation on theme: "2-5: Techniques for Evaluating Limits"— Presentation transcript:

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

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:


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