2. 4.4 Indeterminate Forms and L’Hôpital’s Rule

3. We want to be able to find the limit of the following functions: However, the methods we learned in Chapter 2 do not work. Because of this, problem a is called an indeterminate form of type 0/0 and problem b is called an indeterminate form of  / . We need to use another method. 4.4 Indeterminate Forms and L’Hôpital’s Rule 2Dr. Erickson

4. 4.4 Indeterminate Forms and L’Hôpital’s Rule 3Dr. Erickson

5. Suppose f and g are differentiable and g′(x)≠0 on an open interval I that contains a (except possibly at a). Suppose that (In other words, we have an indeterminate form) Then, If the limit on the right side exists or is ∞or -∞. 4.4 Indeterminate Forms and L’Hôpital’s Rule 4Dr. Erickson

6.  Note 1: L’Hôpital’s Rule says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives, provided that the given conditions are satisfied. It is especially important to verify the conditions regarding the limits of f and g before using l’Hôpital’s Rule.  Note 2: L’Hôpital’s Rule is also valid for one-sided limits and for limits at infinity or negative infinity; that is, “x  a” can be replaced by any of the symbols x  a +, x  a –, x , or x  – .  Note 3: Sometimes L’Hôpital is spelled L’Hospital. Just be aware that the “s” is silent! 4.4 Indeterminate Forms and L’Hôpital’s Rule 5Dr. Erickson

7. Find 4.4 Indeterminate Forms and L’Hôpital’s Rule 6Dr. Erickson

8. 4.4 Indeterminate Forms and L’Hôpital’s Rule 7Dr. Erickson

9. Find the limit. Use l’Hôpital’s Rule where appropriate. 4.4 Indeterminate Forms and L’Hôpital’s Rule 8Dr. Erickson

10. 4.4 Indeterminate Forms and L’Hôpital’s Rule 9Dr. Erickson

11. Find the limit. Use l’Hôpital’s Rule where appropriate. 4.4 Indeterminate Forms and L’Hôpital’s Rule 10Dr. Erickson

12.  Several indeterminate forms arise from the limit: Options: 1. and Type: 0 0 2. and Type:  0 3. and Type: 1  4.4 Indeterminate Forms and L’Hôpital’s Rule 11Dr. Erickson

13.  In each of the three cases, we need to take the natural logarithm of both sides.  NOTE: 0  is NOT indeterminate! 4.4 Indeterminate Forms and L’Hôpital’s Rule 12Dr. Erickson

14. Find the limit. Use l’Hôpital’s Rule where appropriate. 4.4 Indeterminate Forms and L’Hôpital’s Rule 13Dr. Erickson

15. Find the limit. Use l’Hôpital’s Rule where appropriate. 4.4 Indeterminate Forms and L’Hôpital’s Rule 14Dr. Erickson