We evaluate by rewriting the function

Since the function is in the form of a geometric sequence with a common ratio   whose absolute value is less than one, or

the limit goes to zero.

First, express the numerator and denominator in terms of sine and cosine, since these two functions are fundamental to trigonometry and are therefore easier to work with:

...by the double angle formula for sine:

Now we can substitute  into the function and determine the limit:

To find , the notation of the minus sign indicates that we want to find the limit as the graph approaches  from the left.  

The right side of the curve of tangent points upwards towards positive infinity.

The answer is:  

To evaluate the limit, we must first pull out a factor consisting of the highest degree term divided by itself (so we are unchanging the function):

After the factor we pulled out goes to 1, and all of the negative exponent terms go to zero (for they have infinity in their denominator, which equals zero), we are left with 

To evaluate the limit, we must first factor out a term consisting of the highest power term divided by itself (so we are unchanging the function inside the limit):

The term we factored out becomes 1, and all of the negative exponent terms as x goes to infinity go to zero (infinity in a denominator makes the whole term go to zero). We are left with

Step 1: Define what kind of function we have here..

We have a rational function..

Step 2: In general, if the value in the denominator is bigger than the numerator, the value of that fraction is getting close to 

Step 3: Plug in  into our function:





The limit of  as  is zero ()

Although this limit looks complex, we simply need to evaluate the function at the limit value.

To evaluate the limit as n goes to infinity, we must factor out a term consisting of the highest power term divided by itself (which is equal to one, so we are not technically changing the original function):

The factor we created goes to one, and the two terms in the denominator both equal zero as n goes to infinity (they are each terms with n in the denominator of a fraction, the denominator going to infinity, which equals zero), so the function itself has zero in the denominator, which equals .

To evaluate the limit, we must first see what side of the function we are evaluating the limit from. The plus sign exponent on 4 indicates we are approaching 4 from the right, using numbers slightly larger than 4. 

This corresponds to the second part of the piecewise function. When we evaluate the limit, . 

When we evaluate the limit using normal methods (substitution), we find that  is reached, which is an indeterminate form. To evaluate the limit, we can use L'Hopital's Rule, which states that when an indeterminate form is reached, 

Applying this to our limit, we get

The derivatives were found using the following rules:

,