# For what values of x, if any, does f(x) = 1/((12x+4)sin(pi+(6pi)/x)  have vertical asymptotes?

Jan 26, 2017

Vertical asymptotes : x = uarr-1/3; $\uparrow \pm 6 , \pm 3 , \pm 2 , \pm 1.5 , \ldots , \pm \frac{6}{n} , \ldots \downarrow$

Horizontal asymptote : $\leftarrow y = 0 \rightarrow$

#### Explanation:

The asymptotes are given by

$\left(3 x + 1\right) \sin \left(6 \frac{\pi}{x}\right) = 0$. So,

x= -1/3 ; x = 6/n, n=+-1, +-2, +-3, ... give asymptotes.

The horizontal asymptote is revealed by

$x \to \pm \infty$, as y to 0#.

The horizontal space between consecutive vertical asymptotes

diminishes from $3 \to 0$, as $x \to {0}_{\pm}$. The vertical end-

asymptotes $x = \pm 6$ are marked, in the first graph

You can study the second graph, for shape near the exclusive

asymptote $x = - \frac{1}{3}$.

I have used ad hoc ( for the purpose ) scales, for clarity.

graph{(4y(3x+1)sin(6pi/x)+1)(x-6-.01y)(x+6+.01y)=0 [-16, 16, -.5, .5]}

graph{(4y(3x+1)sin(6pi/x)+1)(x+.333-.00001y)=0 [-.4 -.0,-10, 10]}