Step 1: To find vertical asymptotes of a rational function, we set the denominator to . We do this because a fraction is undefined with the denominator is , therefore an asymptote has been calculated..



Factor:



Set each parentheses equal to  and find :




Step 2: We said in Step 1 that any values that we got after setting the denominator to zero would be the asymptotes..

So, the asymptotes of  are .

It can also be written as .

Vertical asymptotes exist where the function divides by zero.  Therefore, it would be tempting to say there is a vertical asymptote at   However, we can factor the numerator of this function (difference of squares), which cancels out the bottom factor.  Therefore, there is no vertical asymptotes of this function.  At , the function has a hole.

Vertical asymptotes occur when the denominator of a function equals zero.  Therefore, we need to solve to find when that happens. 

Remember, whenever we have quadratic solutions, there are always positive and negative roots.

Horizontal asymptotes occur when the degree of the polynomial matches between the numerator and denominator or when the degree of the denominator polynomial is less than the numerator degree.  Here, we have the former case.  The horizontal asymptote in this case is the ratio of the coefficients in front of the leading polynomial.

Since we are looking for a horizontal asymptote, this refers to a  value.

Step 1: To find the horizontal asymptote, imagine what happens to the graph when x is very big. 



Step 2: After taking the limit of the function as x approaches infinity, we can see that the graph will eventually give me 

The line,  is also known as the x-axis.

 is defined as the limit of the function as  approaches  from the right.

To do so, we approximate values slightly larger than 

...

Following this pattern we see that

 is defined as the limit of the function as  approaches four from both sides.

To do so we evaluate the left and right limit, and if they are equal, then that is the limit.

can be approximated by values of  slightly less than  to find that

can be approximated by values of  slightly greater than  to find that

Because the two limits are not equal,

When starting with limits, our first step is always direct substitute the value into the equation.  Unfortunately, when you do that, you get zero in the denominator.  Since we cannot divide by zero, we must look for a method to simplify the function.  We can!  The numerator is factorable.  In this case, we can remove the discontinuity, since it is a hole.

For infinity limits, we do not need to consider any numbers that are being added or subtracted to the functions.  This is because when you plug infinity in for the variables, adding or subtracting another number will not have any effect.  Therefore, our function simplifies to:

Since this is a bottom heavy fraction (the denominator keeps growing, but the numerator does not), this will always tend to zero.

For infinity limits, we only consider parts of the function that have a variable.  Everything else will not matter when plugging in extremely negative values:

Since the exponents are the same on the numerator and denominator, the function tends to a ratio of the leading coefficients.  Moreover, it does not matter if we are going negative or positive infinity since this is just tending to a value without any variables left.