$\lim_{(x,y)\to 0} \frac{x\sin(y)- y\sin(x)}{x^4 + y^4}$ 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?

You can easily disprove that the limit exists by considering the one dimensional family of rays $y=\lambda x, \lambda\in\mathbb{R}$ and taking the limit to the origin along them instead, for a given $\lambda$. If the limit exists then it shouldn't depend on $\lambda$. 
The limit on the rays boils down to evaluating:
$$L=\lim_{x\to 0}\frac{x\sin(\lambda x)-\lambda x\sin x}{(1+\lambda^4)x^4}$$
which one can evaluate straightforwardly using L'Hopital's rule or a Taylor expansion to yield:
$$L=\frac{\lambda-\lambda^3}{6(1+\lambda^4)}$$
which does depend on $\lambda$ and therefore the limit cannot exist.
