# How do you find a vertical asymptote for f(x) = tan(x)?

$f \left(x\right) = \tan x$ has infinitely many vertical asymptotes of the form:
$x = \frac{2 n + 1}{2} \pi$,
We can write $\tan x = \frac{\sin x}{\cos x}$, so there is a vertical asymptote whenever its denominator $\cos x$ is zero. Since
$0 = \cos \left(\frac{\pi}{2}\right) = \cos \left(\frac{\pi}{2} \pm \pi\right) = \cos \left(\frac{\pi}{2} \pm 2 \pi\right) = \cdots$,
$x = \frac{\pi}{2} + n \pi = \frac{2 n + 1}{2} \pi$,
where $n$ is any integer.