# For what values of x, if any, does f(x) = 1/((x-5)sin(pi+(3pi)/x)  have vertical asymptotes?

$x = 5 \mathmr{and} x = \pm 3 / n , n = \pm 1 , \pm 2 , \pm 3 , \ldots$,
$f \left(x\right) = \frac{1}{\sin \left(\frac{3 \pi}{x}\right) \left(5 - x\right)}$, using $\sin \left(\pi + a\right) = - \sin a$
$f \to \infty$, as $x \to$ 3/(an integer) and 5#..