Question

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

Answer 1

Vertical asymptotes : `x = uarr-1/3;` `uarr +-6, +-3, +-2, +-1.5, ...,+-6/n, ...darr`

Horizontal asymptote : `larry = 0rarr`

Explanation:

The asymptotes are given by

`(3x+1)sin(6pi/x)=0`. So,

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

The horizontal asymptote is revealed by

`x to +-oo`, as y to 0#.

The horizontal space between consecutive vertical asymptotes

diminishes from `3 to 0`, as `x to 0_(+-)`. The vertical end-

asymptotes `x = +-6` are marked, in the first graph

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

asymptote `x = -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]}