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