Question

For what values of x, if any, does `f(x) = 1/((x-5)sin(pi+1/x)` have vertical asymptotes?

Answer 1

`1/((x-5)sin(pi+1/x))` has vertical asymptotes at `x=5` and `x=1/(mpi)`

Explanation:

Vertical asymptotes of `1/((x-5)sin(pi+1/x))` are there when

either `x-5=0` i.e. `x=5`

or when `sin(pi+1/x)=0` i.e. `pi+1/x=npi`, where `n` is an integer

or `1/x=pi(n-1)` or `x=1/(pi(n-1))=1/(mpi)`, where `m` is an integer so that `m=n-1`. Note that as `m` increases, value of `x` decreases continuously and maximum value (less than `5`) is `x=1/pi=0.3183`

graph{1/((x-5)sin(pi+1/x)) [-0.996, 0.996, -0.498, 0.498]}

graph{1/((x-5)sin(pi+1/x)) [-14.41, 17.46, -8.86, 7.08]}