Calculus and Analytical Geometry

1. MTH 104 Calculus and Analytical Geometry Lecture # 11

2. L’HÔPITAL’S RULE: INDETERMINATE FORMS INDETERMINATE FORMS OF TYPE . Limit of the form in which and is called an indeterminate form of type

3. L’Hopital Rule for form 0/0 Suppose that and are differentiable functions on an open , and that , except possible at interval containing , then If exists, or if this limit is or Moreover this statement is also true in the case of limits as or as

4. EXAMPLEFind the limit Using L’HÔPITAL’S rule, and check the result by factoring. solution form Using L’HÔPITAL’S rule

5. By computation Example In each part confirm that the limit is an indeterminate form of type 0/0 and evaluate it using L’HOPITAL’s rule (a) (b) (c) (d)

6. (e) (f) Solution (a) Applying L’HÔPITAL’S rule

7. (b) Applying L’HÔPITAL’S rule (c) Applying L’HÔPITAL’S rule

8. (d) Applying L’HÔPITAL’S rule

9. INDETERMINATE FORMS OF TYPE The Limit of a ratio, in which the numerator has limit and the denominator has the limit is called an indetreminate form of type L’Hopital Rule for form Suppose f and g are differentiablefunctions on an open intervalconatining x=a, exceptpossiblyat, x=a and that and , then or exists, or if this limit is If Moreover this statement is also true in the case of limits as or as

10. Example In each part confirm that the limit is indeterminate form of type and apply L’HÔPITAL’S (b) (a) solution (a) Applying L’HÔPITAL’S rule

11. (b) Applying L’HÔPITAL’S rule Any additional application of L’HÔPITAL’S rule will yield powers of in the numerator and expressions involving and in the denominator.

12. Rewriting last expression Thus,

13. INDETERMINATE FORMS OF TYPE The limit of an expression that has one of the forms is called and indeterminate form if the limits and individually exert conflicting influences on the limit of the entire expression. For example Indeterminate form On the other hand Not an indeterminate form

14. Indeterminate form of type can sometimes be evaluated by rewriting the product as a ratio, and then applying L’HÔPITAL’S rule for indeterminate form of type or . Example Evaluate (b) (a) Solution (a) Rewriting form

15. Applying L’HÔPITAL’S rule (b) Rewriting as

16. Applying L’HÔPITAL’S rule

17. Indeterminate forms of type A limit problem that leads to one of the expressions . is called an indeterminate form type The limit problems that lead to one of the expressions are not indeterminate, since two terms work together.

18. Example Evaluate Solution Rewriting Applying L’HÔPITAL’S rule

19. Again Applying L’HÔPITAL’S rule

20. INDETERMINATE FORM OF TYPE Limits of the form can give rise to indeterminate forms of the types and For example It is indeterminate because the expressions and gives --Two conflicting influences. Such inderminate form can be evaluated by first introducing a dependent variable The limit of lny will be an indeterminate form of type

21. Example Solution Let Applying L’HÔPITAL’S rule