Asymptotes: Examples (including the one with the hole)

In general, you will be given a rational (fractional) function, and you will need to find the domain and any asymptotes. You'll need to find the vertical asymptotes, if any, and then figure out whether you've got a horizontal or slant asymptote, and what it is. To make sure you arrive at the correct (and complete) answer, you will need to know what steps to take and how to recognize the different types of asymptotes.

What are the steps for finding asymptotes of rational functions?

Given a rational function (that is, a polynomial fraction) to graph, follow these steps:

1. Set the denominator equal to zero, and solve. The resulting values (if any) tell you where the vertical asymptotes are.
2. Check the degrees of the polynomials for the numerator and denominator.
   • If the denominator is of greater degree, then there is a horizontal asymptote, and it's the x-axis.
   • If the degrees of the numerator and denominator are the same, then there is a horizontal asymptote, and it's the line formed by the ratio of the two leading coefficients. That is, of the numerator's leading coefficient is a and the denominator's leading coefficient is b, then the asymptote is the line .
   • If the degree of the numerator is exactly 1 more than the degree of the denominator, then there is a slant (or oblique) asymptote, and it's found by doing the long division of the numerator by the denominator, yielding a straight (but not horizontal) line.

Now let's get some practice:

• Find the domain and all asymptotes of the following function:

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A given rational function may or may not have a vertical asymptote (depending upon whether the denominator ever equals zero), but (at this level of study) it will always have either a horizontal or else a slant asymptote.

Note, however, that the function will only have one of these two; you will have either a horizontal asymptote or else a slant asymptote, but not both. As soon as you see that you have one of them, don't bother looking for the other one.

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• Find the domain and all asymptotes of the following function:

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What is the special case with the hole?

Sometimes, a rational expression will have a factor that is common to each of the numerator and the denominator. When this factor is cancelled off, the result is a rational expression whose graph has a hole in it.

We've dealt with various sorts of rational functions. When you were first introduced to rational expressions, you likely learned how to simplify them. You'd factor the polynomials top and bottom, if you could, and then you'd see if anything cancelled off.

What if you've found the zeroes of the denominator of a rational function (so you've found the spots disallowed in the domain), but one or another of the factors cancels off? If there is a factor that's common to the numerator and denominator, then simplifying means that you'll cancel off that factor — and thus delete one of the zeroes of the denominator. But surely zeroes can't just disappear, right?

Let's look at an example of exactly that situation:

• Find the domain and all asymptotes of the following function:

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This last case (being "the case with the hole") is not the norm for slant asymptotes, but you should expect to see at least one problem of this type, including perhaps on the test.

By the way, when you go to graph the function in this last example, you can draw the line right on the slant asymptote. But you will need to leave a nice open dot (indicating the hole) where x = 2, to indicate that this point is not actually included in the graph because it's not part of the domain of the original rational function.

The only hard part is remembering that sometimes a factor from the denominator might cancel off, thereby removing a vertical asymptote but not changing the restrictions on the domain. You might even want to get in the habit of checking if the polynomials in the numerator and denominator factor, just in case.

Either way, when you're working these problems, try to go through the steps in order, so you can remember the whole process on the test. These exercises are not so hard once you get the hang of them, so be sure to do plenty of practice exercises.