Evaluating Limits Analytically Lesson 2.3

2 What Is the Squeeze Theorem? Today we look at various properties of limits, including the Squeeze Theorem

3 Basic Properties and Rules  Constant rule  Limit of x rule  Scalar multiple rule  Sum rule (the limit of a sum is the sum of the limits) See other properties pg

4 Limits of Functions  Limit of a polynomial P(x) Can be demonstrated using the basic properties and rules  Similarly, note the limit of a rational function What stipulation must be made concerning D(x)?

5 Try It Out  Evaluate the limits Justify steps using properties

6 General Strategies

7 Some Examples  Consider Why is this difficult?  Strategy: simplify the algebraic fraction

8 Reinforce Your Conclusion  Graph the Function Trace value close to specified point  Use a table to evaluate close to the point in question

9 Some Examples  Rationalize the numerator of rational expression with radicals  Note possibilities for piecewise defined functions

10 Three Special Limits  Try it out! View Graph View Graph View Graph View Graph View Graph View Graph

11 Squeeze Rule  Given g(x) ≤ f(x) ≤ h(x) on an open interval containing c And … Then

12 Assignment  Lesson 2.3A  Page 87  Exercises 1-43 odd  Lesson 2.3B  Page 88  Exercises 45 – 97 EOO