# How do you find the Vertical, Horizontal, and Oblique Asymptote given f(x) = 7 tan(πx)?

Apr 5, 2018

See below.

#### Explanation:

There are no oblique asymptotes. Oblique asymptotes occur in rational functions where the numerator is of higher degree then the denominator. This is not a rational function. The function is periodic, so no horizontal asymptotes.

Vertical asymptotes occur where the function is undefined.

$\tan x$ is undefined at $\frac{\pi}{2}$ , $\pi + \frac{\pi}{2}$ etc.

We can express this as:

$n \pi + \frac{\pi}{2}$ where n is an integer.

$\therefore$

$y = 7 \tan \left(\pi x\right)$

$\pi x = \frac{\pi}{2} + n \pi$

$x = \frac{1}{2} + n$

so the function is undefined when $x = \frac{1}{2} + n$

$n \in \mathbb{Z}$

GRAPH: