Question

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

Answer 1

By looking at the graph, you can see repeated vertical asymptotes starting at `pi/2` and repeating every `pi` units.

Explanation:

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 `pi/2`. The opposite side will get larger and larger as the adjacent side gets smaller and smaller. At `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.

As you go around the unit circle, this pattern repeats, leading to the repeated asymptotes you see. In Quadrant 3, the same thing happens but it goes to negative infinity, still a vertical asymptote. In quadrants 2 and 4 the reverse happens, leading to a very small number divided by a very large number, which is why the graph goes through the x-axis (y=0) between each set of asymptotes.