We explore functions that “shoot to infinity” at certain points in their domain.

If we think of an asymptote as a “line that a function resembles when the input or output is large,” then there are three types of asymptotes, just as there are three types of lines:

Here we’ve made up a new term “slant” line, meaning a line whose slope is neither zero, nor is it undefined. Let’s do a quick review of the different types of asymptotes:

Vertical asymptotes Recall, a function has a vertical asymptote at if at least one of the following hold:

  • ,
  • ,
  • .

In this case, the asymptote is the vertical line

Horizontal asymptotes We have also seen that a function has a horizontal asymptote if and in this case, the asymptote is the horizontal line

Slant asymptotes On the other hand, a slant asymptote is a somewhat different beast.

Consider the graph of the following function. What is the slant asymptote of this function?

To analytically find slant asymptotes, one must find the required information to determine a line:

  • The slope.
  • The -intercept.

While there are several ways to do this, we will give a method that is fairly general.