# For what values of x, if any, does #f(x) = 1/((x-2)sin(pi+(8pi)/x) # have vertical asymptotes?

##### 1 Answer

This function has a vertical asymptote for

#### Explanation:

A rational function has a vertical asymptote if its denominator equals zero. In your case, the denominator is

As every product, it equals zero if and only if at least one of its factors equals zero.

It is easy to find out that the first factor equals zero if

As for the second, we can first of all note that

So, we need to find out when

This function has an infinite number of solutions, given by

So, for