Question

How do I find the asymptotes of `y=x/sin(x^2)`? Are there even any asymptotes?

Answer 1

Vertical asymptotes : `uarrx=+-sqrt(kpi)darr, k = 0, 1, 2, 3, ...`.

See Socratic graph, for highlights.

Explanation:

`y =(1/x)(x^2/(sinx^2)`

The second factor has an indeterminate form `0/0`, at x = 0.

As `x to 0, y to lim1/x lim(x^2/sin(x^2))=+-oo(1)=+-oo`.

Besides this vertical asymptote x = 0, the others are given by the

zeros `x =+-sqrt(kpi), k = 1. 2. 3, ...`, of `sinx^2`.

Socratic asymptotes-inclusive graph, for `x in [-pi, pi]`#, is included.

graph{(y-x/sin(x^2))=0 [-20, 20, -10, 10]}
4
graph{(x+.004y)(x-1.77+.005y)(x+1.77+.005y)(y-x/sin(x^2))(x-2.54-.003y)(x+2.54+.003y)=0 [-3.1416 3.1416 -10, 10]}

Ad hoc scales used to highlight asymptotes relative graph, for `x in (-pi, pi)`.