1. Calculus and Analytical Geometry
MTH 104
Lecture # 11
Calculus and Analytical Geometry

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. EXAMPLE Find 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 differentiable functions on an open interval conatining x=a, except possibly at, x=a and that
and
, then
exists, or if this limit is or 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
(a)
(b)
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