# How do you find the vertical asymptotes of f(x) = tan(πx)?

May 2, 2018

The vertical asymptotes occur whenever $x = k + \frac{1}{2} , k \in \mathbb{Z}$.

#### Explanation:

The vertical asymptotes of the tangent function and the values of $x$ for which it is undefined.

We know that $\tan \left(\theta\right)$ is undefined whenever $\theta = \left(k + \frac{1}{2}\right) \pi , k \in \mathbb{Z}$.

Therefore, $\tan \left(\pi x\right)$ is undefined whenever $\pi x = \left(k + \frac{1}{2}\right) \pi , k \in \mathbb{Z}$, or $x = k + \frac{1}{2} , k \in \mathbb{Z}$.

Thus, the vertical asymptotes are $x = k + \frac{1}{2} , k \in \mathbb{Z}$.

You can see more clearly in this graph:

graph{(y-tan(pix))=0 [-10, 10, -5, 5]}