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**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)

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**Basic Limits (on handout)**

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**Properties of Limits (on handout)**

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**Properties of Limits (on handout)**

(n is even) (n is odd)

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

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**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

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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 xc is not f(c) if f(x) is discontinuous at c. ↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑ Cannot use substitution!! Must be other methods

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**There is no hope… Important Idea Horrible Occurrence!!!**

If substitution results in an a/0 fraction where a0, the limit doesn’t exist. There is no hope… Horrible Occurrence!!!

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**There is HOPE!! 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 the factor and cancellation technique Substitution doesn’t work…does this mean the limit doesn’t exist? Try substitution 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 Try substitution**

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

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**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

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Try This Find the limit if it exists:

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**Try This Find the limit if it exists: Confirm by graphing**

The limit doesn’t exist

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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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**Horizontal Asymptotes!!**

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 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…

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Try This Find the limit if it exists:

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Limits and Their Properties 11.2 Copyright © Cengage Learning. All rights reserved.

Limits and Their Properties 11.2 Copyright © Cengage Learning. All rights reserved.

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