# What are the asymptote(s) and hole(s), if any, of  f(x) = tanx?

Dec 9, 2017

$f \left(x\right) = \tan \left(x\right)$ is a continuous function on its domain, with vertical asymptotes at $x = \frac{\pi}{2} + n \pi$ for any integer $n$.

#### Explanation:

$f \left(x\right) = \tan \left(x\right)$

has vertical asymptotes for any $x$ of the form $x = \frac{\pi}{2} + n \pi$ where $n$ is an integer.

The value of the function is undefined at each of these values of $x$.

Apart from these asymptotes, $\tan \left(x\right)$ is continuous. So formally speaking $\tan \left(x\right)$ is a continuous function with domain:

$\mathbb{R} \text{\} \left\{x : x = \frac{\pi}{2} + n \pi , n \in \mathbb{Z}\right\}$

graph{tan x [-10, 10, -5, 5]}