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Let's think about straight lines for a minute. As you know, most straight lines can be described by the equation \(y = mx + b\). Vertical lines are the exception and we write them as \(x = a\), where \(a\) is a constant.

The three types of asymptotes we talk about on this page cover all possibilities of straight lines. Vertical asymptotes are vertical lines. Horizontal asymptotes are lines with slope zero written as \(y = b\). Slant asymptotes are all other lines, \(y = mx + b\).

We can extract these equations of lines from rational functions. That is what we discuss on this page.

As outlined on the basics of rational functions page, vertical asymptotes occur in rational functions at x-values where the denominator is zero AND the numerator is NOT zero. This implies that even though we have zero in the denominator of the rational function and, therefore, the x-value is not in the domain, we still need to look at the numerator at that x-value. If the numerator is not zero then we have a vertical asymptote at that x-value. (If the numerator IS zero, then there is a hole there. We discuss that on the basics of rational functions page.) Here is a graph and it's corresponding equation showing an example of a vertical asymptote. This graph has an asymptote at \(x = 3\).

\(\displaystyle{ f(x)=\frac{1}{x-3} }\)
Vertical Asymptote at \(x = 3 \)

In the case of a vertical asymptote, it doesn't matter if the graph goes off to positive or negative infinity. The x-value where the denominator is zero is called an asymptote in both cases. Also notice in this graph that one side goes off to positive infinity, the side goes off to negative infinity. Sometimes both sides go off in the same direction, positive or negative. For our discussion, the direction and whether or not they are the same does not matter. (You get to consider thoses cases in calculus.) In all cases, they are just called vertical asymptotes.

Horizontal and slant asymptotes cover the other two cases of \(y = mx + b\). In these cases, we look at what happens to the y-value as x gets very large or very small.
Before we get too far into this discussion, make sure you remember how to do long division of polynomials (covered on a separate page). It will also help you to review synthetic division, since it is faster than long division. Synthetic division works in fewer cases but, when can be used, it will save time.

These asymptotes occur as x goes to positive infinity or negative infinity, i.e. as x gets very small or very large. To determine these asymptotes, we follow these steps.

Before we go on, let's watch a couple of videos discussing horizontal asymptotes. Many teachers teach these concepts but without giving you the full context that horizontal asymptotes are just one of several types of these asymptotes. So keep that in mind as you watch these videos. She shows some great examples here too.

A couple of quick examples will help. In these examples, we have already separated the rational function using long division. So the remaining part that is still a rational function has a smaller order numerator than denominator.

Example 1 - \(\displaystyle{f(x)=1+\frac{1}{x}}\)

Example 1 - - What is the asymptote of \(\displaystyle{f(x)=1+\frac{1}{x}}\)?
As x gets larger and larger, \(1/x\) gets smaller and smaller and eventually will become zero. This leaves only \(y=1\) for very large x. The same argument holds as x moves in the negative direction. So \(y=1\) is the asymptote, a horizontal asymptote in fact.

Example 2 - \(\displaystyle{f(x)=x+3+\frac{1}{x}}\)

Example 2 - - What is the asymptote of \(\displaystyle{f(x)=x+3+\frac{1}{x}}\)?
Using the same logic as in example 1, the asymptote is \(y=x+3\), in this case a slant asymptote.

Similar results hold for other types of problems. The key is to use long division to get the form we need.

Okay, time for the practice problems.