# How do you find a vertical asymptote for a rational function?

Let $f \left(x\right) = \frac{p \left(x\right)}{q \left(x\right)}$ be a rational function. A line $x = {x}_{0}$ is a vertical asymptote of $f$ when
${\lim}_{x \to {x}_{0}^{\pm}} \frac{p \left(x\right)}{q \left(x\right)} = \pm \infty .$
Since a rational function is continuous in its domain, the possible vertical asymptote $x = {x}_{0}$ are among that for which $q \left({x}_{0}\right) = 0$.
In other words, first we have to find a point ${x}_{0}$ that is not in the domain of $f$, ie, $q \left({x}_{0}\right) = 0$, and then verify if limits of $f$ are $\pm \infty$ when x goes to ${x}_{0}^{+}$ and ${x}_{0}^{-}$.