# For what values of x, if any, does f(x) = -tan(pi/12-x)  have vertical asymptotes?

Jan 23, 2018

See below.

#### Explanation:

We recognize that, as:

as $\theta \to \frac{\pi}{2} \textcolor{w h i t e}{88888}$ , $\tan \left(\theta\right) \to \infty$ ( Undefined )

The function will have vertical asymptotes everywhere it is undefined. i.e. when:

$\left(\frac{\pi}{12} - x\right) = \frac{\pi}{2}$

Solving for $x$:

$x = - \frac{\pi}{2} + \frac{\pi}{12} = - \frac{5 \pi}{12}$

This will also be true if we add integer multiples of $\pi$ onto this:

$- \frac{5 \pi}{12} + n \pi$

For $\textcolor{w h i t e}{8888} n \in \mathbb{Z}$

GRAPH: