Figure

EXAMPLE 2Finding a Vertical Asymptote

Find any vertical asymptote(s) of the graph of $f(x)=\dfrac{x}{(x-3)^{2}}$ .

The domain of $f$ is $\{x|x\neq 3\}$ . Since $3$ is the only number for which the denominator of $f$ equals zero, we construct Table 13 and investigate the one-sided limits of $f$ as $x$ approaches $3$ . Table 13 suggests that $\lim\limits_{x\rightarrow 3}\dfrac{x}{(x-3)^{2}}=\infty$

So, $x=3$ is a vertical asymptote of the graph of $f$ .

Figure 52

$f(x)=\dfrac{x}{(x-3)^2}$

TABLE 13

$\underrightarrow{x~\hbox{approaches 3 from the left}}$							$\underleftarrow{x~\hbox{approaches 3 from the right}}$
$x$	$2.9$	$2.99$	$2.999$	$\ \rightarrow$	3	$\leftarrow$	$3.001$	$3.01$	$3.1$
$f( x) =\dfrac{x}{(x-3)^{2}}$	$290$	$29{,}900$	$2{,}999{,}000$	$f(x)$ becomes unbounded			$3{,}001{,}000$	$30{,}100$	$310$

120

Figure 52 shows the graph of $f( x) =\dfrac{x}{( x-3)^{2}}$ and its vertical asymptote.