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

##### 1 Answer
Jan 2, 2017

Period: $\frac{\pi}{9}$. For each period $\left(\frac{\pi}{12} + \frac{k}{9} \pi , \frac{7}{36} \pi + \frac{k}{9} \pi\right)$, there are two terminal asymptotes x = end value, $k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

#### Explanation:

As $\left(\frac{\pi}{4} - 9 x\right) \to$ (an odd multiple ) $\left(2 k + 1\right)$ of $\frac{\pi}{2}$,$f \to \pm \infty$,

giving

$f \to \pm \infty$, as $x \to - \frac{4 k + 1}{36} \pi , k = 0 , \pm 1 , \pm 2 , \pm 3 , . .$

So, in ine period $x \in \left(\frac{\pi}{12} , \frac{7}{36} \pi\right)$, we have two terminal

asymptotes

$x = \frac{\pi}{12} \mathmr{and} x = \frac{7}{36} \pi$

The period for f is $\frac{\pi}{9} = \left(\frac{7}{36} - \frac{1}{12}\right) \pi$.

graph{y-tan(0.7854-9x)=0 [-5, 5, -2.5, 2.5]}