# For what values of x, if any, does f(x) = tanx  have vertical asymptotes?

By looking at the graph, you can see repeated vertical asymptotes starting at $\frac{\pi}{2}$ and repeating every $\pi$ units.
This comes from the definition of tangent, which is the ratio of the opposite side of a right triangle to it's adjacent side. Picture a right triangle as $\theta$ gets closer to $\frac{\pi}{2}$. The opposite side will get larger and larger as the adjacent side gets smaller and smaller. At $\frac{\pi}{2}$ the opposite side will become infinitely large as the adjacent side becomes infinitely small. So opposite divided by adjacent will give you infinity on the graph, which is a vertical asymptote.