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2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover (www.mrgover.com) Objectives: Find limits using direct substitution Find limits when substitution doesnt work

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Basic Limits (on handout)

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Properties of Limits (on handout)

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(n is even) (n is odd)

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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

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

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

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Try This Find the limit, if it exists: 3

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

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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

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Important Idea If substitution results in an a /0 fraction where a 0, the limit doesnt exist. There is no hope… Horrible Occurrence!!!

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

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Example Find the limit if it exists: Try substitution Substitution doesnt work…does this mean the limit doesnt exist? Try the factor and cancellation technique Go to Derive

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

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

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Procedure 1.Try substitution 2. Factor and cancel if substitution doesnt work 3.Try substitution again The factor & cancellation technique

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Try This Find the limit if it exists: 5 Isnt that easy? Did you think calculus was going to be difficult?

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Try This Find the limit if it exists:

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Try This Find the limit if it exists: The limit doesnt exist Confirm by graphing

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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.

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

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Example Find the limit, if it exists:

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Example Find the limit, if it exists:

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Example Find the limit, if it exists:

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Example Find the limit if it exists: Try substitution With substitution, you get an indeterminate form Try factor & cancel Factor & cancel doesnt work Horrible Occurrence!!! The rationalization technique to the rescue… Rationalizing the numerator allows you to factor & cancel and then substitute

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Try This Find the limit if it exists:

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