Find multivariable limit: as $(x,y) \to (0,0)$ of $ \frac {x\sin y-y\sin x} {x^2+y^2}$

Question

How do I evaluate the following limit?

$$ \lim_{(x,y) \to (0,0)} \frac {x\sin y-y\sin x} {x^2+y^2}$$

So far I tested the limit along the paths in which $y=mx$ and I got zero. However I have no clue how I can prove the limit exists or find the limit's value.

Answer 1

$\textbf{Hint:}$

$$\frac{|x\sin y - y\sin x|}{x^2+y^2} = \frac{|xy|}{x^2+y^2}\left|\frac{\sin y}{y}-\frac{\sin x}{x}\right| \leq \frac{1}{2}\left|\frac{\sin y}{y}-\frac{\sin x}{x}\right|$$

By AM-GM inequality.