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Evaluating Limits Analytically

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1. 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. 79-81

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 tospecified point • Use a table to evaluateclose to the point inquestion

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

11. Squeeze Rule • Given g(x) ≤ f(x) ≤ h(x) on an open interval containing cAnd … • Then

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