# For what values of x, if any, does f(x) = 1/((4x+9)cos(pi/2+(4pi)/x)  have vertical asymptotes?

Asymptote is vertical at

$x = - \frac{9}{4} , \frac{4}{k}$

Where $k = 0 , \setminus \pm 1 , \setminus \pm 2 , \setminus \pm 3 , \setminus \ldots$

#### Explanation:

The given function:

$f \left(x\right) = \frac{1}{\left(4 x + 9\right) \setminus \cos \left(\setminus \frac{\pi}{2} + \frac{4 \setminus \pi}{x}\right)}$

$f \left(x\right) = \frac{1}{\left(4 x + 9\right) \setminus \sin \left(\frac{4 \setminus \pi}{x}\right)}$

Above function will have vertical asymptotes when denominator becomes equal to zero i.e.

$\left(4 x + 9\right) \setminus \sin \left(\frac{4 \setminus \pi}{x}\right) = 0$

$4 x + 9 = 0 , \setminus \setminus \mathmr{and} \setminus \setminus \setminus \sin \left(\frac{4 \setminus \pi}{x}\right) = 0$

$x = - \frac{9}{4} , \setminus \setminus \mathmr{and} \setminus \setminus \setminus \frac{4 \setminus \pi}{x} = k \setminus \pi$

$x = - \frac{9}{4} \setminus \setminus \setminus \mathmr{and} \setminus \setminus \setminus x = \frac{4}{k}$

Where, $k$ is any integer hence we get the set of points where asymptote is vertical

$x = - \frac{9}{4} , \frac{4}{k}$

Where $k = 0 , \setminus \pm 1 , \setminus \pm 2 , \setminus \pm 3 , \setminus \ldots$