Horizontal and Slant Asymptotes - At A Glance

Vertical asymptotes are fairly easy to find. Horizontal and slant asymptotes are a bit more complicated, though. Not actually complicated, but they require a little more work. Just warning you ahead of time.

The horizontal asymptote is the value that the rational function approaches as it wings off into the far reaches of the x-axis. It's all about the graph's end behavior as x grows huge either in the positive or the negative direction. The equation of a horizontal asymptote will be "y = some constant number." Just like the equation of any horizontal line. Hint, hint.

We've already seen a few horizontal asymptotes in the previous section, like in our buddy up there.

Horizontal asymptotes aren't nearly as strict as vertical ones. It doesn't matter to them how many times they are crossed over, as long as the graph approaches them in the end.

Now, here's how we find horizontal asymptotes. It's all a matter of degrees. The degree of the polynomial, that is:

• If the degree of the numerator (up top) is smaller than the degree of the denominator (down below), then the horizontal asymptote is the x-axis itself (y = 0).

• If the degree of the numerator (top dog) is equal to the degree of the denominator (down low Joe), then we look at the leading coefficient of each polynomial. Divide them out, and the horizontal asymptote is y = that number.

Let's take one final look at . Each polynomial has degree 1. So, at that point we ignore every term except the leading coefficients. We have , and as expected, our graph has a horizontal asymptote at y = 1.

Here's another one, .

The numerator has degree 1, while the denominator has degree 2. As x gets larger and larger—let's say 1000—check out the first term of each polynomial. We have . We have to squint to see a number that tiny, and it's going to get even smaller as x gets larger. That's why having a bottom-heavy rational expression leads to a horizontal asymptote at y = 0.

Front-ways, Side-ways, and Slant-ways

What will happen when a rational function is top-heavy by 1 in its degree? Will it tip and tumble downhill? No, even worse: it forces us to use long division.

That's because a rational function with the degree of the numerator exactly 1 larger than the degree of the denominator has a slant asymptote. It's an asymptote that follows some line, y = f(x). We find the line by going through the long division of our polynomial fraction.

Sample Problem

Find the equation of any asymptotes of .

We've got to go through long division here.

Our division has a remainder, but we don't care about that. Throw it out like week-old food. The equation of our slant asymptote is y = x + 4.

Don't go celebrating just yet, though. We also need to look for vertical asymptotes and holes. Yes, we can have multiple types of asymptotes, plus points of discontinuity. Don't freak out over it; just remember to check for all of them. Factoring the numerator, we get:

Nothing cancels out, so we just have a vertical asymptote at x = 1.

See how the line approaches but never touches the line y = x + 4? Slant asymptotes are just as touchy as vertical ones. Poke at your own peril.

For any rational function, we need to check for horizontal and slant asymptotes by scoping out the degree of each polynomial. Don't be a creeper, though; just ask to take a look. Unlike vertical asymptotes and holes, we will only find one horizontal or one slant asymptote.