Through direct substitution, we see that the limit becomes

which is in indeterminate form.

As such we can use l'Hospital's Rule, which states that if the limit 

is in indeterminate form, then the limit is equivalent to

Taking the derivatives we use the trigonometric rule which states 

where  is a constant.

Using l'Hospital's Rule we obtain

And through direct substitution we find

As such the limit exists and is

When evaluting the limit using normal methods, we find that the indeterminate form  is reached. When this (or ) happens, we use L'Hopital's Rule to evaluate the limit:

So, we must find the derivative of the numerator and denominator:

When we plug these into the formula and evaluate the limit we get:

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

.

This problem requires the product rule. The derivative using the product rule is as follows:

Let  and .

Their derivatives are:

 and 

Substitute the functions into the product rule formula and simplify.