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

Feb 16, 2017

Vertical asymptotes : $\uparrow x = \pm \sqrt{k \pi} \downarrow , k = 0 , 1 , 2 , 3 , \ldots$.

See Socratic graph, for highlights.

#### Explanation:

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

The second factor has an indeterminate form $\frac{0}{0}$, at x = 0.

As $x \to 0 , y \to \lim \frac{1}{x} \lim \left({x}^{2} / \sin \left({x}^{2}\right)\right) = \pm \infty \left(1\right) = \pm \infty$.

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

zeros $x = \pm \sqrt{k \pi} , k = 1. 2. 3 , \ldots$, of $\sin {x}^{2}$.

Socratic asymptotes-inclusive graph, for $x \in \left[- \pi , \pi\right]$#, 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 $x \in \left(- \pi , \pi\right)$.