Classify the discontinuity of tan(x) at x = pi/2

Since $a(x) = \dfrac{\sin(x)}{\cos(x)}$, and $\cos\left(\frac{\pi}{2}\right) = 0$ while $\sin\left(\frac{\pi}{2}\right) = 1 \neq 0$, the function is undefined at $x = \frac{\pi}{2}$ because we'd be dividing by zero. This alone rules out $a(x)$ being continuous there, but we need to look at the one-sided behavior to classify the discontinuity.

As $x \to \frac{\pi}{2}^-$, $\cos(x) \to 0^+$, so $a(x) \to +\infty$; as $x \to \frac{\pi}{2}^+$, $\cos(x) \to 0^-$, so $a(x) \to -\infty$. Since at least one (in fact both) one-sided limits are infinite, this corresponds to a vertical asymptote at $x = \frac{\pi}{2}$, so the discontinuity is classified as infinite.