Hello everyone, I have the following function:

$\displaystyle \frac{(sen(xy))}{\sqrt{x^2+y^2}} \rightarrow if (x,y) \neq (0,0) $

$\displaystyle 0 \rightarrow if (x,y) = (0,0)$

I have to find out its continuity and differentiability in (0,0).

Concerning the continuity:

$\displaystyle \lim_{(x,y)\rightarrow (0,0)} \frac{(sen(xy))}{\sqrt{x^2+y^2}} \rightarrow \frac{(sen(xy))}{\sqrt{x^2+y^2}} * \frac{xy}{xy} \rightarrow \frac{(sen(xy))}{xy} \frac{xy}{\sqrt{x^2+y^2}}$

senxy/xy goes to 1. While the other one:

$\displaystyle \lim_{(x,y)\rightarrow (0,0)} \frac{xy}{\sqrt{x^2+y^2}} \leqslant \frac{|xy|}{\sqrt{x^2+y^2}} \leqslant \frac{1}{2} \frac{x^2+y^2}{\sqrt{x^2+y^2}} \Rightarrow \lim_{(x,y)\rightarrow (0,0)} \frac{1}{2}x^2+y^2 = 0$

This is what I did so far, is this correct? If it isn't, why? Concerning the differentiability I'll tell how I did as soon as I understand if I did this right (because if I did this wrong is useless to post the other part!). Thanks a lot!