# What are the asymptotes for f(x) = tan(2x)?

Note that $\tan \left(2 x\right) = \sin \frac{2 x}{\cos} \left(2 x\right)$.
This expression will form asymptotes when $\cos \left(2 x\right) = 0$.
This expression is zero when $2 x = \pm \left(n + \frac{1}{2}\right) \pi , n = 0 , 1 , 2. . .$.
Therefore the asymptotes occur at $x = \pm \frac{1}{2} \left(n + \frac{1}{2}\right) \pi , n = 0 , 1 , 2. . .$