Vertical asymptotes.

Definition 4.7.1   Let be a function defined on a (pointed) neighborhood of , where . If has an infinite (one-sided) limit at , then the line whose equation is is an asymptote to the graph of .

Example 4.7.2   Let . As and , the axis is an asymptote to the graph of (namely an hyperbola). Actually each one of these one-sided limits is enough to ensure the existence of the asymptote.

Example 4.7.3   Let . This function is defined over and is periodic with period . Moreover the function is odd (please check this!). Let us compute the limits of at the open ends of half a period, say .

Hence the lines whose respective equations are (the axis) and are vertical asymptotes to the graph of .

As is odd, we have that the line whose equation is is a an asymptote to ; by periodicity we conclude that all the lines whose equations are , for integer , are asymptotes to . Part of the graph of this function is displayed in Figure 11.

At this point, we wish to mention a very common error: it is not sufficient for a function not to be defined at a single point for the line whose equation is to be an asymptote to the graph of . For example, let be given by . Then , and the axis is not an asymptote to the graph of . See Figure 12.

We said already that an infinite one-sided limit at shows the existence of a vertical asymptote whose equation is . In example function discussion rational function of exp we will see a situation where the function has two different one-sided limits at one point: the one is finite, the other is infinite, showing the existence of a vertical asymptote.

Noah Dana-Picard 2007-12-28