Question

How to find vertical asymptotes of cotangent function?

Answer 1

Vertical asypmtotes are line of the form $x = k$, where k is a point not in the domain of the function.

Since the definition of the cotangent function is

$\cot(x) = \dfrac{\cos(x)}{\sin(x)}$

The function is not defined where the denominator is zero, i.e.

$\sin(x) = 0 \iff x = k\pi,\ k \in \mathbb{Z}$

So, every line with equation $x = k\pi$ is a vertical asymptote for the cotangent function. Some examples may be

$x = 0,\ x = \pi,\ x = -3\pi,\ x = 152\pi,\ x = -234\pi,\ldots$