Asymptote

An asymptote of a graph is an imaginary line that the graph gets close to as x goes to plus or minus infinity or a particular number. For example, the graph of the rational function f(x) = 1x has two asymptotes — the x-axis and the y-axis.

Analyzing the diagram, the following can be observed.

• As x approaches infinity and as x approaches negative infinity, the value of the function approaches 0. Therefore, y = 0, or the x-axis, is a horizontal asymptote of the graph of f.
• As x approaches 0, the value of the function approaches either positive or negative infinity. Therefore, x = 0, or the y-axis, is a vertical asymptote of the graph of f.

In the coordinate plane below, the asymptotes for three different graphs are shown.

The applet demonstrates that asymptotes can be not only vertical and horizontal, but also oblique.