# For what values of x, if any, does f(x) = -tan(pi/6-x)  have vertical asymptotes?

$f \left(x\right)$ have Vertical Asymptotes for $x = - \frac{\pi}{3} \pm n \cdot 180$

where $n = 0 , 1 , 2 , 3. \ldots .$

#### Explanation:

From the given $f \left(x\right) = - \tan \left(\frac{\pi}{6} - x\right)$

Set the angle $\left(\frac{\pi}{6} - x\right) = \frac{\pi}{2}$ because $\tan \left(\frac{\pi}{2}\right) =$undefined

Solving for $x$

$x = - \frac{\pi}{3}$

and for tangent function the values are the same everytime $\pi$ or multiples of $\pi$ are added or subtracted.
Therefore, the function has asymptotes for values of
$x = \frac{\pi}{3} \pm n \cdot \pi$ where $n = 0 , 1 , 2 , 3 , \ldots . .$
Examples:

$x = - \frac{\pi}{3}$

$x = - \frac{4 \pi}{3}$

$x = \frac{2 \pi}{3}$

$x = \frac{5 \pi}{3}$