Determine $\lim_{(x,y)\to(0,0)}\frac{\sin(xy)}{(x^4+y^4)}$ Determine $$\lim_{(x,y)\to(0,0)}\frac{\sin(xy)}{(x^4+y^4)}$$
Hint:
For small enough $x,y$ such that the ordered pair $(x,y)\ne (0,0)$, we have $|\sin(xy^2)|\lt |xy^2|$
$|y|=\sqrt{y^2}\le\sqrt{y^2+x^2}$ and so by similar arguments, we have: 
$\big|\frac{x^2 y\sin(xy^2)}{(x^4+y^4)\sqrt{x^2+y^2}}-0\big|\le \frac{|x^3|y^2}{x^4+y^4}\le \frac{|x|x^2}{\sqrt{x^4+y^4}}\le |x|\le \sqrt{x^2+y^2}$
Can you fill in the gaps and finish using definition of limit?
