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**EVALUATING LIMITS ANALYTICALLY**

Section 1.3

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**When you are done with your homework, you should be able to…**

Evaluate a Limit Using Properties of Limits Develop and Use a Strategy for Finding Limits Evaluate a Limit Using Dividing Out and Rationalizing Techniques Evaluate a Limit Using the Squeeze Theorem

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SOME BASIC LIMITS Let and b and c be real numbers and let n be a positive integer.

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Evaluate 5 -3 Does not exist

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Evaluate 1 -1 5 Does not exist

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Properties of Limits Let and b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: 1. Scalar multiple Sum or difference Product Quotient 5. Power

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Evaluate -4 -2 1 Does not exist

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**Limits of Polynomial and Rational Functions**

If p is a polynomial function and c is a real number, then If r is a rational function given by and c is a real number such that then

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Evaluate 5 -5 Does not exist

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**Evaluate the function at x = 2**

1 DNE

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**The Limit of a Function Involving a Radical**

Let n be a positive integer. The following limit is valid for all c if is n odd, and is valid for if n is even.

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**The Limit of a Composite Function**

Let f and g be functions with the following limits: Then

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**Limits of Trigonometric Functions**

Let c be a real number in the domain of the given trigonometric function.

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**STRATEGIES FOR FINDING LIMITS**

Functions That Agree at All But One Point Let c be a real number and let for all x in an open interval containing c. If the limit of g as x approaches c exists, then the limit of f also exists and

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**A Strategy for Finding Limits**

Learn to recognize which limits can be evaluated by direct substitution. If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to find a function g that agrees with f for all x other than x = c. Apply . Use a graph or table to reinforce your conclusion.

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**Dividing Out Techniques**

Example:

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Evaluate. Does not exist 1/16

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**Rationalizing Techniques**

Example:

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**Evaluate the exact limit.**

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**TWO SPECIAL TRIGONOMETRIC LIMITS**

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**Evaluate the exact answer.**

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Warm-Up Thoughts(after 1.2) Why do piece-wise functions and rational functions make for great “limit” examples and discussion. Think of at least 3 reasons.

Warm-Up Thoughts(after 1.2) Why do piece-wise functions and rational functions make for great “limit” examples and discussion. Think of at least 3 reasons.

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