Question

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

Answer 1

This function has a vertical asymptote for `x=2` and for every other `x` of the form `8/k`, for `k \in \mathbb{Z}`

Explanation:

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

# (x-2)\sin(pi+(8pi)/x) #

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 `x=2`.

As for the second, we can first of all note that `sin(\pi+z)=-\sin(z)`

So, we need to find out when

# -\sin((8pi)/x)=0 \iff \sin((8pi)/x)=0 #

This function has an infinite number of solutions, given by

# (8pi)/x=k\pi \iff 8/x=k \iff x=8/k #

So, for `x=...-8/4,-8/3,-8/2,-8,8,8/2,8/3,8/4...` the denominator equals zero, and the function has a vertical asymptote.