# For what values of x, if any, does f(x) = tan((3pi)/4-9x)  have vertical asymptotes?

Oct 11, 2017

$x = - \frac{\pi}{36}$ and $x = - \frac{\pi}{18}$ for $0 \le \theta \le \left(2 \pi\right)$

#### Explanation:

Vertical asymptotes occur at values for which the function is undefined. Using this, we know that:

$0 \le \theta \le \left(2 \pi\right)$

as $\theta \to \pi , \tan \left(\theta\right) \to \infty$

also as $\theta \to \frac{3 \pi}{2} , \tan \left(\theta\right) \to - \infty$

This means that if:

$\left(\frac{3 \pi}{4} - 9 x\right) = \pi \mathmr{and} \frac{3 \pi}{2}$

The function will be undefined, and vertical asymptotes will occur.

So:

$\frac{3 \pi}{4} - 9 x = \pi \implies x = - \frac{\pi}{36}$

and:

$\frac{3 \pi}{4} - 9 x = \frac{3 \pi}{2} \implies x = - \frac{\pi}{18}$

Graph of $f \left(x\right) = \tan \left(\frac{3 \pi}{4} - 9 x\right)$