Limit of the function $\lim \limits_{(x,y)\to (0,0)} \sin (\frac{x^2}{x+y}) \ (x+y \neq 0)$. Since $\sin x$ is a continuous function at $(0,0)$ it suffices to check if the limit $\lim_{(x,y)\to (0,0)} \frac{x^2}{x+y}$ is finite. 
I seem to be missing the idea in order to show that the limit $\lim_{(x,y)\to (0,0)} \frac{x^2}{x+y}=0$. I tried converting to polar form I get $\lim_{r\to 0} \frac{r \cos^2 \theta}{\cos \theta +\sin \theta}$. It seems that this is not conclusive. Any help?

In fact, along the curve $y=x^3-x,$ $\sin(x^2/(y+x)) = \sin(1/x)$ takes on every value in $[-1,1]$ infinitely many times in any deleted neighborhood of $(0,0).$
