I would like to show that:
$ \frac{x\sin(y)-y\sin(x)}{x^2+y^2}\rightarrow_{(x,y)\to (0,0)}0$
$ \left| \frac{x\sin(y)-y\sin(x)}{x^2+y^2} \right| \leq \frac{2\vert xy \vert}{x^2+y^2} \leq 1$ which is not sharp enough, obviously.
How can I efficiently "dominate" the quantity $ \vert x\sin(y)-y\sin(x)\vert$ ?