<s>[INST] The continuity of $f(x,y) = \frac {\sin (xy)} {\sqrt {x^2 + y^2}}$ at $(0,0)$ 
I got a function: 
$f(x,y) = \frac {\sin (xy)}  {\sqrt {x^2 + y^2}}$ if $(x,y) \not =
 (0,0)$, otherwise $f(x,y) = 0$,

and I should look at its continuity in $(0,0)$


*

*I tried the sandwich theorem but got slightly stuck unfortunately
$ 0 \le \left|\frac {\sin (xy)}  {\sqrt {x^2 + y^2}}\right| \le 1$
I was thinking that $\sin (xy)$ is a restricted function, but not sure what to with $\sqrt {x^2 + y^2}$ if $x,y$ are too small so I think my idea doesn't work. Then it would look like:
$ 0 \le \left| {\sin (xy)} \right| \le \sqrt {x^2 + y^2} = 0$
Any other idea please?
 [/INST] We have
$|sinxy|\leq xy$. Since $|x|\leq \sqrt{x^2+y^2}$ and $|y|\leq \sqrt{x^2+y^2}$ then
$|sinxy|\leq xy\leq x^2+y^2$
then 
$|\frac{sinxy}{\sqrt{x^2+y^2}}|\leq \frac {xy}{\sqrt{x^2+y^2}}\leq\frac{ x^2+y^2}{\sqrt{x^2+y^2}}\leq\frac{\delta^2}{\delta}\leq \delta$
So, it is sufficiently put $\epsilon=\delta$.
</s>