As we have with previous functions we have studied, let’s look at the end behavior of rational functions. Recall that we noticed that the function approached 0 as x approached infinity or negative infinity. In general, if the -values of a function approach a specific number as approaches infinity or negative infinity, we call that a horizontal asympote of the function.

Horizontal Asymptotes

This means that as gets bigger and bigger (in the positive or negative direction), the -values of the function get close to a particular value . Anther way to understand this that as gets bigger and bigger (in the positive or negative direction), the graph of the function approaches the horizontal line .

We know from looking at that some rational functions have horizontal asymptotes. The question we would like to investigate is which rational functions have horizontal asymptotes, and if a rational function does have a horizontal asymptote, how do we determine it’s value? Let’s look at a few particular examples.

Now, you must be asking yourself if every time we want to test for vertical or horizontal asymptotes, we need to keep plugging values and guessing. Fortunately, the answer is “no”. With a little bit of algebra, it becomes much more apparent what the end behavior of a rational function will be. Formal justificatives require Calculus — we’ll be content with getting intuition for now.

We will need the following theorem.

Hopefully this theorem seems intuitive. If goes to 0 as gets big, then raising to a higher power should just make the function get close to zero more quickly. If we take this theorem as a given, we can use it and some algebra to rewrite other rational functions so that we can find their horizontal asymptotes.

The technical reason why this works require calculus, but for now it is enough to know that if you can rewrite a rational function so it contains terms of , then as goes to infinity or negative infinity, those terms will become . Let’s try another example.

You may have noticed a pattern. In both examples above, we just ended up with the coefficients of the leading terms from the top and bottom of the rational function. This pattern holds whenever the top and bottom fractions have the same degree. The theorem below gives the pattern for horizontal functions in general.

The above theorem essentially says that one can detect horizontal asymptotes by looking at degrees and leading coefficients. Only the leading terms of and matter, and it makes no difference whether one considers or . For example, how would you apply this to study horizontal asymptotes for the following function?

Slant asymptotes

From the theorem above, we can see that in the third case when the degree of the numerator is bigger than the degree of the denominator, there are no horizontal asymptotes.. It would be natural to ask whether we can still say anything about the rational function in that case. It turns out that we can, and the key is to use long division of polynomials. Whenever the degree of the numerator is bigger that the degree of the denominator, we can use long division to rewrite the function as a polynomials plus a rational function where the degree of the numerator is less than the degree of the denominator.

Note that saying that is a slant asymptote for the graph of is the same thing as saying that is a horizontal asymptote for the graph of the difference function .

The above examples suggest that if the degree of the numerator is at least two higher than the degree of the numerator, what survives outside the remainder has degree higher than one, and thus does not describe a line equation — meaning no slant asymptotes. Similarly, if the degree of the numerator is equal or lower to the degree of the denominator, there’s “not enough quotient left” to describe a line equation. This is not a coincidence, but a general fact.

Unlike what happened for horizontal and vertical asymptotes, the above theorem does not immediately tell you what is the line equation describing the slant asymptote. We must resort to long division.