1.3 Evaluating Limits Analytically

Warm-up Find the roots of each of the following polynomials

Properties of Limits Theorem 1.1 Let b and c be real numbers and let n be a positive integer

Theorem 1.2 Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits

Examples

Theorem 1.3 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 r(x) = p(x)/q(x) and c is a real number such that q(c) = 0, then

Theorem 1.4 The Limit of a Functions Involving a Radical Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even

Theorem 1.5 The Limit of a Composite Function Theorem 1.6 Limits of Trigonometric Functions

A Strategy for Finding Limits Check to see if the function can be evaluated by direct substitution If direct substitution yields an indeterminate form (0/0), try using the dividing out or rationalizing techniques

Dividing Out

Rationalizing

Answers to last night’s homework Pi / /3 7.4/315. DNE 8.0

Another type of indeterminate form occurs when you take the limit of a function and your answer is in the form b/0 Ex.

Finding Limits of Trig Functions Use the unit circle given to you to complete the following problems

Theorem 1.8 The Squeeze Theorem If h(x) f(x) g(x) for all x in an open interval containing c, except possibly at c itself, and if:

Theorem 1.9 Two Special Trig Limits Ex.

Homework: p.65-67: 17 – 21 odd, 23, 25, 39, 41, 43, 49 – 61 odd, 67 – 73 odd, 103, 104.