« Limits of Trigonometric Functions

L'Hôpital's Rule for 

Suppose . Then

If , then .

If      tends to  in the limit, then so does .

Here is a case of    .

Examples 1:

Examples 2:

Examples 3:

Example4:

Sometimes it is necessary to use L'Hôpital's Rule several times in the same problem.

Example5:

Here is a more elaborate example involving the indeterminate form 0/0. Applying the rule a single time still results in an indeterminate form. In this case, the limit may be evaluated by applying L'Hôpital's rule three times:

L'Hôpital's Rule for 

Suppose . Then

If , then .

If     tends to  in the limit, then so does .

Here is a case of   

Example 6:

Example 7:

Iterate the above until the exponent is 0. Then one sees that the limit is 0.

Example 8:

This one also involves :

Basic indeterminate forms (all others reduce to these):

Other indeterminate forms:

Example 9:

To handle a case of  , the difference of two functions is converted to a quotient:

Example 9:

To handle a case of , the difference of two functions is converted to a quotient:

When NOT to use l'Hôpital's rule

because