Indeterminate Forms and L’Hopital’s Rule Chapter 4.4 April 12, 2007

Method for Improper Integrals: Define: Evaluate the Integral and then evaluate the limit If there is an infinite integral in both directions, then we define the integral to be:

Evaluate: Let the variable “n” be so that the interval [1,n] is finite. Looking at the limit as n goes to infinity: We need a better way to calculate the limits…….

We ended with the improper integral where we needed to evaluate the limit: Let’s review evaluating limits with a look at the following examples:

Indeterminate Forms Suppose we want to find. If we substitute 0 in for x, we get. The limit may or may not exist. It’s indeterminate. (We can’t determine the limit.) Any limit of the form is called an indeterminate form of type. We call the form an indeterminate form of type. Other Indeterminate forms are:

L’Hopital’s Rule If the Or Then Take for example: Indeterminate form: Going back to our improper integral: Provided the limit on the right exists (or is or )

Our example fits the criteria for L’Hopital’s Rule: Let the variable “n” be so that the interval [1,n] is finite. Looking at the limit as n goes to infinity: L’Hopital’s Rule applied to the first limit. We say that the integral converges to 1.

Back to our beginning examples L’Hopital’s does not apply! L’Hopital’s DOES apply! Which is the same answer as before using algebra.

More Examples: = 1/2, L’Hopital’s does NOT apply!

More Examples: Is finite. Determine a value of b for which b= 32/3

More Examples: The form is: L’Hopitals does not apply, but if we rewrite the limit as the resulting form will be or And we can apply L’Hopital’s Rule.

More Examples: The form is: Rewrite the limit as the resulting form will be And we can apply L’Hopital’s Rule.