Continuity of $f(x,y)=\dfrac{(xy+\sqrt{x^2+y^2})\sin^2 (x+y)}{x^2+y^2}$ 
$f(x,y)=\dfrac{(xy+\sqrt{x^2+y^2})\sin^2 (x+y)}{x^2+y^2}$ if $xy\neq0$ and $f(x,y)=0$ if $xy=0$, check whether $f(x,y)$ is continous at origin.

For it to be continuous $$\lim_{(x,y)\rightarrow (0,0)} f(x,y)=0$$
using epsilon-delta definition,$$\begin{align}\left|\dfrac{(xy+\sqrt{x^2+y^2})\sin^2 (x+y)}{x^2+y^2}\right|&\\\leq\dfrac{xy+\sqrt{x^2+y^2}}{x^2+y^2}&\\\leq\left|\dfrac{xy}{x^2+y^2}\right|+\left|\dfrac{1}{\sqrt{x^2+y^2}}\right|&\\\leq\dfrac{1}{2}+\left|\dfrac{1}{\sqrt{x^2+y^2}}\right|\end{align}$$
So I'm unable to put this in delta does limit DNE? But if I draw its graph, it looks to me continuous.
Please help.

$\sin^{2}(x+y)\leq (x+y)^{2} \leq 2(x^{2}+y^{2})$. Hence $|f(x,y)|\leq 2|xy|+2\sqrt {x^{2}+y^{2}} \to 0$. 
