1. 1.3 Evaluating Limits Analytically, continued! Rationalizing Technique, Squeeze Theorem (AKA The Sandwich Theorem)

2. Like Ex 8 p. 64 Find the limit: Direct substitution gives . . . We need to find equivalent expression! How? Check out graph!

3. Great, clear as mud, right? Gotta find me a trap!

4. The squeeze theorem will help prove Theorem 1.9 Instead of using x as the variable, to avoid confusion in the proof, we will use θ

6. Multiply all by 2/sin θ and you get: Take reciprocals and reverse inequalities: Look at limits of outside functions, cos θ and 1. As x approaches 0, each has a limit of 1. Then apply Squeeze theorem Middle expression is what we are trying to find limit of as x approaches 0

7. Like Ex 9 p.66 Find the limit: How to start? Like Ex 10 p.66 Find the limit: I need the same thing in the denominator as I have inside the sine function. How do I get that form?

8. Assignment: 1.3 p. 68/ 53, 56, 57-69 odd, 75-79 odd, 91, 97 (14 probs)