Question

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

Answer 1

`x=2`, `x=2pik`, and `x=pi+2pik` 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:
`(x-2)sinx=0`
`x-2=0` and `sinx=0`
`x=2` and `x=2pik, pi+2pik` where `k in Z` (`k` is an integer)

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