1.3 Evaluating Limits Analytically

1. 1.3 Evaluating Limits Analytically Objectives: -Students will evaluate a limit using properties of limits -Students will develop and use a strategy for finding limits -Students will evaluate a limit using dividing out and rationalizing techniques -Students will evaluate a limit using the Squeeze Theorem

2. Properties of Limits

3. Ex 1)

4. Ex 2) Yesterday we found that but we had to do this graphically; direct substitution didn’t work because of the hole (the denom was 0) To find the limit analytically…simplify! Find the same function but without the hole.

5. Strategy for finding : • Direct substitution • Simplify to an identical function except at x=c ; then use direct substitution. • Use a graph or table to check. • Remember, sometimes the limit DNE!!

6. Simplifying Techniques: 1) Factoring (cancellation)

7. Simplifying Techniques: 2) Rationalization (rationalize numerator by multiplying by conjugate over itself)

8. Simplifying Techniques: 3) ∆x problems → expand then reduce

9. Simplifying Techniques: 4) Common denominator

10. Other Things to Look For: Squeeze Theorem -if for all x in an open interval containing c, except possibly at c, and if , then

11. Two Special Trigonometric Limits: ** memorize these!!

12. Ex 3)

13. Ex 4) → check using graphs or tables!

14. Ex 5)