« Limits of Trigonometric Functions 

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'Hospital'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:
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):
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'Hospital's rule
because