Evaluating Limits Analytically Lesson 2.3

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

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

What stipulation must be made concerning D(x)? 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)?

Try It Out Evaluate the limits Justify steps using properties

General Strategies

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

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

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

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

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

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