# For what values of x, if any, does f(x) = 1/((x-2)sinx)  have vertical asymptotes?

Mar 13, 2016

$x = 2$, $x = 2 \pi k$, and $x = \pi + 2 \pi k$ where $k \in Z$

#### Explanation:

Vertical asymptotes occur whenever the denominator equals 0. To find them, we simply set the denominator to 0 and solve, like thus:
$\left(x - 2\right) \sin x = 0$
$x - 2 = 0$ and $\sin x = 0$
$x = 2$ and $x = 2 \pi k , \pi + 2 \pi k$ where $k \in Z$ ($k$ is an integer)

We have infinitely many vertical asymptotes, at $x = 2$, $x = 2 \pi k$, and $x = \pi + 2 \pi k$. We can see this on the graph of the function below: