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

Dec 26, 2016

$\frac{2}{3} \mathmr{and} \frac{8}{k - 1} , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots ,$

#### Explanation:

As $y \to 0 , x \to \pm \infty$.

As $x \to {\left(\frac{2}{3}\right)}_{+} , y \to \pm \infty$.

Also, for example, as $x \to {\left(\frac{8}{k - 1}\right)}_{+} , k = - 3 , y \to \pm - \infty$

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]}