Determine $\lim_{(x,y)\to(0,0)}\frac{\sin(xy)}{(x^4+y^4)}$ Determine $$\lim_{(x,y)\to(0,0)}\frac{\sin(xy)}{(x^4+y^4)}$$
Since $|\sin t| \leq |t|$ for all $t\in\mathbb{R}$,
$$
|x^2 y \sin(xy^2)| \leq |x|^3 |y|^3 \leq \frac{x^6+y^6}{2}\,.
$$
the second inequality by the AM-GM inequality. This means
$$
\frac{|x^2 y \sin(xy^2)|}{(x^4+y^4)\sqrt{x^2+y^2}}
\leq \frac{1}{2} \frac{(x^6+y^6)}{(x^4+y^4)\sqrt{x^2+y^2}}
\leq \frac{1}{2}\frac{2\max(x^6,y^6)}{\max(x^4,y^4)\sqrt{\max(x^2,y^2)}}
= \max(|x|,|y|)
$$
and $\lim_{(x,y)\to (0,0)} \max(|x|,|y|) = 0$, so you can conclude by the squeeze theorem.
