1. Indeterminate Forms and L’Hopital’s Rule Part 2 Chapter 4.4 April 17, 2007

2. Indeterminate Forms 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:

3. L’Hopital’s Rule applies ONLY to indeterminate forms: If the Or Then Take for example: Indeterminate form: Provided the limit on the right exists (or is or )

4. However we can manipulate expressions of other forms so they fit the criteria: 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.

5. However we can manipulate expressions of other forms so they fit the criteria: The form is: L’Hopitals does not apply, but if we rewrite the limit as a quotient (finding a common denominator) the resulting form will be or And we can apply L’Hopital’s Rule.

6. Example: The form is: L’Hopitals does not apply, but we can combine the log term using properties of logs: Which gives us the form

7. The last forms involve exponents and we’ll use these properties of logs: in our solution. The form is: L’Hopitals does not apply, but if we rewrite the function in terms of e using the first property of logs: We can use the second property and find the limit using our previous forms.

8. More Examples:

9. Try:

10. More Examples: Is finite. Determine a value of b for which b= 0 Determine a value of b for which Is finite.b= 8 Determine the behavior of the limits: Both diverge