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

1 Answer
Sep 6, 2014

#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.