Continuity of $f(x,y) = \frac{x}{y}\sin(x^2+y^2)$ if $y\neq 0$,$ f(x,0) = 0$

Question

This function is continuous in $c=(x,y)$ when $y\neq0$ and it is not continuous when $y=0$ and $\sin x^2 \neq 0$. I think it is also not continuous when $y=0$ and $\sin(x^2) = 0$ , particularly at $(0,0)$. However, in this case, I can´t find a subset for which the limit is different from $0$. I know there are other ways to prove this, but it would be much easier if I could find a proper set. Can you help me?

Answer 1

hint

$$f(x,y)=\frac xy\sin(x^2+y^2)=$$ $$x\sin(x^2)\frac{\cos(y^2)}{y}+x\frac{\sin(y^2)}{y}\cos(x^2)$$

$$f(x,x^3)\to 1$$ and $$f(x,-x^3)\to -1$$