# For what values of x, if any, does f(x) = 1/((x^2-4)cos(pi/2+(7pi)/x)  have vertical asymptotes?

Feb 3, 2017

$x = \pm 2 , \pm 7 \left(1 , \frac{1}{2} , \frac{1}{3} , \frac{1}{4} , \ldots\right)$

#### Explanation:

The vertical asymptotes are given by

zero of ${x}^{2} - 4 = 0$ and $\cos \left(\frac{\pi}{2} + \frac{7}{x} \pi\right) = 0 \implies - \sin \left(7 \frac{\pi}{x}\right)$.

These are

$x = \pm 2$ and $\frac{7}{k} , k = \pm 1 , \pm 2 , \pm 3 , \ldots$

Note that k = 0 sends $x \to \pm \infty$ to make kx read 7 .

The ad hoc graph is not to scale.
graph{-1/((x^2-4)sin(21.991/x) [-3.51, 3.47, -17, 17]}