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)

Basic Limits (on handout)

Properties of Limits (on handout)

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

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

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

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

Try This Find the limit, if it exists: 3

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

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 

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!!!

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

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

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

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

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

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

Try This Find the limit if it exists:

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

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.

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

Example Find the limit, if it exists:

Example Find the limit, if it exists:

Example Find the limit, if it exists:

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…

Try This Find the limit if it exists: