# For what values of x, if any, does f(x) = 1/x-tan(x)  have vertical asymptotes?

At $x = 0$ and at odd multiples of $\frac{\pi}{2}$
$f \left(x\right) = \frac{1}{x} - \tan x = \frac{1}{x} - \sin \frac{x}{\cos} x = \frac{\cos x - x \sin x}{x \cos x}$
The denominator is $0$ and the numerator is not at $x = 0$ and at odd multiples of $\frac{\pi}{2}$.
So there are vertical asymptotes at those value of $x$.