Question

For what values of x, if any, does `f(x) = 1/((3x-2)sin(pi+(8pi)/x)` have vertical asymptotes?

Answer 1

`2/3 and 8/(k-1), k = 0, +-1, +-2, +-3, ...,`

Explanation:

As `y to 0, x to +-oo`.

As `x to (2/3)_+, y to +-oo`.

Also, for example, as `x to (8/(k-1))_+, k = -3, y to+--oo`

All these asymptotes are displayed in the Socratic graph.

graph{y(y(3x-2)sin(3.1418(1+8/x))-1)(3x-2)(x+2)=0 [-3.1, 3.118, -1.446, 1.662]}