Indeterminate Forms and L'Hospital's Rule

THEOREM (L'Hospital's Rule): Suppose and are differentiable and near (except possibly at ). Suppose that



or that



(In other words, we have an indeterminate form of type or ) Then



if the limit on the right side exists (or is or ).


Indeterminate Forms of Type and

EXAMPLES:

1. Find


2. Find


3. Find


4. Find


5. Find


6. Find



Common Mistakes

7. Find


8. Find


9. Find



Indeterminate Forms of Type and

EXAMPLES:

10. Find


11. Find


12. Find



Indeterminate Forms of Type and
13. Find


14. Find


15. Find


16. Find


17. Find


18. Find


COMPARE: We have



since



Similarly,



since




19. Find



Appendix A

THEOREM: The function is differentiable and






Appendix B

1. Find


2. Find


3. Find


4. Find