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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. 1. 2. 3.

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Evaluate A.5 B.-3 C.Does not exist

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Evaluate A.1 B.-1 C.5 D.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 2.Sum or difference 3.Product 4.Quotient 5.Power

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Evaluate A.-4 B.-2 C.1 D.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 A.5 B.0 C.-5 D.Does not exist

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Evaluate the function at x = 2 A.1 B.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. A.0 B.Does not exist C.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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