<s>[INST] Prove that $\lim_{(x,y)\to (0,0)}\frac{x^{2}+xy+y^{2}}{\sqrt{x^{2}+y^{2}}}=0$. 
Prove that $\displaystyle \lim_{(x,y)\to (0,0)}\frac{x^{2}+xy+y^{2}}{\sqrt{x^{2}+y^{2}}}=0$.

This has been my rough work so far, but I am not sure how to go further or if I am doing it the wrong way...
$\displaystyle|f(x,y)-L|=\left |\frac{x^{2}+xy+y^{2}}{\sqrt{x^{2}+y^{2}}}  \right | \leq \frac{|x^{2}+xy+y^{2}|}{\big|\sqrt{x^{2}+y^{2}}\big|}\cdot \frac{\sqrt{x^{2}+y^{2}}}{\sqrt{x^{2}+y^{2}}} = \frac{\big(\sqrt{x^{2}+y^{2}}\big)|x^{2}+xy+y^{2}|}{|x^{2}+y^{2}|}$
I wanted the square root on top to help determine what I should set my $\delta$ as. Any ideas on how to finish this or do it in a better way?
 [/INST] You may also procced as follows using GM-QM (inequality between geometric and quadratic mean):


*

*$\sqrt{ab}\leq \sqrt{\frac{a^2+b^2}{2}}$
So, you get 
\begin{eqnarray*}  \left| \frac{x^2+xy+y^2}{\sqrt{x^2+y^2}}\right|
 & = & \left| \frac{x^2+y^2}{\sqrt{x^2+y^2}} + \frac{xy}{\sqrt{x^2+y^2}}\right| \\
 & \leq  &   \sqrt{x^2+y^2} + \frac{|xy|}{\sqrt{x^2+y^2}}\\
 & \stackrel{GM-QM}{\leq} & \sqrt{x^2+y^2} + \frac{\left(\sqrt{\frac{x^2+y^2}{2}} \right)^2}{\sqrt{x^2+y^2}} \\
& = & \frac{3}{2}\sqrt{x^2+y^2} \\
& \stackrel{(x,y)\to (0,0)}{\longrightarrow} & 0
\end{eqnarray*}
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