We being by attempted to plug in  into our given function.

Since this would yield , we can use L'Hospital's Rule to help us find the limit.

Replace the numerator and the denominator of our function with their respective derivatives, and we get

Hence the answer is .

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  again yields the indeterminate form

So we apply L'Hospital's rule again

Evaluating for  yields

As such

and thus

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

When we evaluate the limit using normal methods, we arrive at the indeterminate form . When this occurs, to evaluate the limit, we must use L'Hopital's Rule, which states that

So, we must find the derivative of the top and bottom functions:

The derivatives were found using the following rule:

Now, rewrite the limit and evaluate it:

When we evaluate the limit using normal methods, we get the indeterminate form . When this happens, we must use L'Hopital's Rule, which states that 

Now, we must find the derivatives of the numerator and denominator:

The derivatives were found using the following rules:

, , 

Next, rewrite the limit and evaluate it:

The first thing we always have to do is to check that l'Hopital's rule is actually applicable when we want to use it.

So it is applicable here.

We take the derivative of the top and bottom, and get

and now we can safely plug in x=1 and get that the limit equals

.

Plugging  into the function  head on yields the inteterminate form of zero times negative infinity, so we must rewrite the problem

. Start

In this expression, when  approaches  from the positive side, the limit "approaches " So we can use L'Hospital's rule.

When we evaluate the limit using normal methods, we get the indeterminate form .

So, we must use L'Hopital's Rule to evaluate the limit, which states that

Using the above, we get

when we evaluate the limit using substitution.

If you plug in the limit value, the function turns into .  Therefore, we are allowed to use l'Hospital's Rule.  We start by taking the derivative of both the numerator and the denominator until when you plug in the value of the limit, you do not get something in the form .  Luckily, in this case, we only need to take the derivative once.  By taking the derivatives separately, we get a new limit: 

.