Vertical asymptotes.

Example 4.7.2 Let $f(x)=\frac 1x$ . As $\underset{x \underset{<}{\rightarrow} 0}{\lim} f(x) = -\infty$ and $\underset{x \underset{>}{\rightarrow} 0}{\lim} f(x) = +\infty$ , 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.

**Figure 10:** The axis as a vertical asymptote.
$\begin{figure}\mbox{\epsfig{file=hyperbola.eps,height=5cm}}\end{figure}$

Example 4.7.3 Let $f(x)=\frac {\cos 2x}{\sin x}$ . This function is defined over $\mathbb{R}-\{ k \pi, k \in \mathbb{Z} \}$ and is periodic with period $2\pi$ . Moreover the function is odd (please check this!). Let us compute the limits of

at the open ends of half a period, say $(0, \pi)$ .

$\begin{align}\begin{cases}\underset{x \underset{<}{\rightarrow} 0 } \cos 2x = 1 \underset{x \underset{<}{\rightarrow} 0 } \sin x =0^- \end{cases}\end{align}$	$\displaystyle \Longrightarrow \underset{x \underset{<}{\rightarrow} 0 } f(x)=-\infty.$
$\begin{align}\begin{cases}\underset{x \underset{>}{\rightarrow} \pi } \cos 2x =... ...\underset{x \underset{<}{\rightarrow} \pi } \sin x = 0^+ \end{cases}\end{align}$	$\displaystyle \Longrightarrow \underset{x \underset{<}{\rightarrow} \pi } f(x)=+\infty.$

Hence the lines whose respective equations are

(the

axis) and $x=\pi$ are vertical asymptotes to the graph $\mathcal{C}$ of

As is odd, we have that the line whose equation is $x=-\pi$ is a an asymptote to $\mathcal{C}$ ; by periodicity we conclude that all the lines whose equations are $x=k \pi$ , for integer , are asymptotes to $\mathcal{C}$ . Part of the graph of this function is displayed in Figure 11.

**Figure 11:** A graph with infinitely many vertical asymptotes.
$\begin{figure}\centering \mbox{ \epsfig{file=VertAsympt-Cos2x_Sinx.eps,height=5cm} }\end{figure}$

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 $f(x)=x \; \ln x$ . Then $\underset{x \underset{>}{\rightarrow} 0}{\lim} f(x)=0$ , and the

axis is not an asymptote to the graph of

. See Figure 12.

**Figure:** The graph of $f: \; x \mapsto x \ln x$ .
$\begin{figure}\centering \mbox{\epsfig{file=xLnx.eps,height=3.5cm}}\end{figure}$

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.