By substituting the value of , we will find that this will give us the indeterminate form .  This means that we can use L'Hopital's rule to solve this problem.

L'Hopital states that we can take the limit of the fraction of the derivative of the numerator over the derivative of the denominator.  L'Hopital's rule can be repeated as long as we have an indeterminate form after every substitution.

Take the derivative of the numerator.

Take the derivative of the numerator.

Rewrite the limit and use substitution.

The limit is .

Directly evaluating for  yields the indeterminate form

we are able to apply L'Hospital's rule which states that if the limit is in indeterminate form when evaluated, then

As such the limit in the problem becomes

Evaluating for  yields

As such

and thus

This important limit from elementary limit theory is usually proven using trigonometric arguments, but it can be shown using L'hopital's rule too.

When evaluating the limit using normal methods, we find that the indeterminate form  results. When this occurs, we must use L'Hopital's Rule, which states that for .

Taking the derivative of the top and bottom functions and evaluating the limit, we get

The derivatives were found using the following rules:

, , 

When evaluating the limit using normal methods, we find that we get the indeterminate form . When this occurs, we must use L'Hopital's Rule to evaluate the limit. The rule states that for the limits for which the indeterminate forms result,

Taking the derivatives for our limit, we get

The derivatives were found using the following rules:

, 

When evaluating the limit using normal methods (substitution), we get the indeterminate form . When this (or ) occurs, we must use L'Hopital's Rule to evaluate the limit. The rule states that when an indeterminate form results from the limit, 

Taking the derivatives in our limit, we get

The derivatives were found using the following rules:

, ,  , 

Admittedly a more computationally intensive problem, we begin by trying to plug in  into the limit directly, but we get the indeterminate form .

We then apply L'Hôpital's rule and try again:

.

However, plugging in  still yields the indeterminate form , so we are forced to apply L'Hôpital's rule a second time:

.

This time, plugging in  does not produce an indeterminate form, so we may evaluate the limit by setting  as follows:

.

Simply plugging in  into the expression yields the indeterminate form of , so we must resort to using L'Hôpital's rule. We take the derivative of the numerator and denominator, and then look at the limit again.

This time, when we plug in , we do not get an indeterminate form, so we can evaluate the limit by setting :

.