Question

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

Answer 1

`f(x)=tan x` has infinitely many vertical asymptotes of the form:
`x=(2n+1)/2pi`,
where n is any integer.

We can write `tan x={sin x}/{cos x}`, so there is a vertical asymptote whenever its denominator `cos x` is zero. Since
`0=cos(pi/2)=cos(pi/2 pm pi)=cos(pi/2 pm 2pi)=cdots`,
we have vertical asymptotes of the form
`x=pi/2+npi={2n+1}/2pi`,
where `n` is any integer.