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
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
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.