1. 1.3 Evaluating Limits Analytically

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

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

4. 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

5. Examples

6. 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

7. 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

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

9. 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

10. Dividing Out

11. Rationalizing

12. Answers to last night’s homework 1.129. 2 2.Pi10. 8 3.811. 12 4.5412. 0 5.-113. -4/5 6.1114. 4/3 7.4/315. DNE 8.0

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

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

15. 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:

16. Theorem 1.9 Two Special Trig Limits Ex.

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