Vertical asymptotes.

    

Definition 4.7.1   Let $ f$ be a function defined on a (pointed) neighborhood of $ x_0$ , where $ x_0 \in \mathbb{R}$ . If $ f$ has an infinite (one-sided) limit at $ x_0$ , then the line whose equation is $ x=x_0$ is an asymptote to the graph of $ f$ .

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 $ y-$ axis is an asymptote to the graph of $ f$ (namely an hyperbola). Actually each one of these one-sided limits is enough to ensure the existence of the asymptote.

Figure 10: The $ y-$ 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 $ f$ 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 $ x=0$ (the $ y-$ axis) and $ x=\pi$ are vertical asymptotes to the graph $ \mathcal{C}$ of $ f$ .

As $ f$ 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 $ k$ , 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.
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\epsfig{file=VertAsympt-Cos2x_Sinx.eps,height=5cm}
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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 $ x_0$ for the line whose equation is $ x=x_0$ to be an asymptote to the graph of $ f$ . For example, let $ f$ be given by $ f(x)=x \; \ln x$ . Then $ \underset{x \underset{>}{\rightarrow} 0}{\lim}
f(x)=0$ , and the $ y-$ axis is not an asymptote to the graph of $ f$ . 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 $ x_0$ shows the existence of a vertical asymptote whose equation is $ x=x_0$ . 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