Evaluating Limits Analytically Chapter 2.5

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)

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)?

Limit of a horizontal asymptote  

General Strategies Verify the limit exist (graphically) Substitute in the value to find the limit. If you get an indeterminate form, try to cancel out a term in the numerator and denominator. Multiply by the conjugate of a term.

Try It Out Evaluate the limits by substitution   4

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

Squeeze Rule (more to say about this later) Given g(x) ≤ f(x) ≤ h(x) on an open interval containing c And … Then

Some Examples