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. 

For infinity limits, we only consider the leading exponents (with their respective coefficients).  Therefore, our complicated functions simplify to the following:

When we plug in more and more negative values, this will keep getting bigger, and the negative signs will cancel each other out.  Therefore, this will tend to positive infinity.

For infinity limits, we need only consider the leading exponents.  Everything else will not matter when we plug in negative infinity. Therefore, our complex limit becomes much simpler:

Because this is an odd exponent and we are plugging in increasingly more negative values, the function will tend to larger and larger negative values. 

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. The horizontal asymptote in this case is the  axis.

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

Look at a graph of .

At , there is a vertical asymptote.

The graph goes down to  as .