$\lim_{(x,y)\to 0} \frac{x\sin(y)- y\sin(x)}{x^4 + y^4}$ without polar coordinates? I have the following limit: 
$$\lim_{(x,y)\to 0} \frac{x\sin(y)- y\sin(x)}{x^4 + y^4}$$
And I must evaluate it without polar coordinates. I have tried a lot of stuff but nothing works. Can someone give me a hint?

The limit does not exist. Take limit along $y=2x$ for example. You get $\lim \frac {2x\sin \, x(\cos \, x -1)} {(1+2^{4})x^{4}}$.  Use the fact that the $\frac {\sin\, x} x \to 1$ and use L'Hopital's Rule to see that the limit is $-1/17$. Since the limit along $x=y$ is $0$ it follows that the given limit does not exist. 
