EVALUATING LIMITS ANALYTICALLY Section 1.3

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

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

Evaluate 5 -3 Does not exist

Evaluate 1 -1 5 Does not exist

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

Evaluate -4 -2 1 Does not exist

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

Evaluate 5 -5 Does not exist

Evaluate the function at x = 2 1 DNE

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.

The Limit of a Composite Function Let f and g be functions with the following limits: Then

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

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 .

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.

Dividing Out Techniques Example:

Evaluate. Does not exist 1/16

Rationalizing Techniques Example:

Evaluate the exact limit. .25 0.0

TWO SPECIAL TRIGONOMETRIC LIMITS

Evaluate the exact answer. 5.0 0.0