Can anyone help me with a hint on how to solve this limit?
$$\lim _{\left(x,\ y\right)\to \left(0,\:0\right)}\frac{\left|x\right|\log \left|x\right|}{\sqrt{\left|x\right|}+\sqrt{\left|y\right|}}$$
I started trying to prove it by paths, but all the limits go to zero. Later, I tried polar coordinates but even using L'Hopital several times I can't get out of indetermination. I also can't think of an inequality to try to use the Squeeze theorem.
Best Answer
As suggested in the comments, since $\sqrt{|x|}+\sqrt{|y|}\ge \sqrt{|x|}$
$$\left| \frac{|x|\log|x|}{\sqrt{|x|}+\sqrt{|y|}} \right|\le \left| \frac{|x|\log|x|}{\sqrt{|x|}} \right| = \sqrt{|x|}\left|\log|x|\right| $$
Can you conclude from here?
As an alternative, by polar coordinates
$$\frac{|x|\log|x|}{\sqrt{|x|}+\sqrt{|y|}}= \frac{\sqrt \rho |\cos \theta|\left(\log \rho+\log|\cos \theta|\right)}{\sqrt{|\cos \theta|}+\sqrt{|\sin \theta|}}\to 0$$
indeed $\sqrt{|\cos \theta|}+\sqrt{|\sin \theta|} \ge \cos^2 \theta+\sin^2 \theta=1$ and
$$\sqrt \rho |\cos \theta|\left(\log \rho+\log|\cos \theta|\right)= \sqrt \rho \log \rho|\cos \theta|+\sqrt \rho|\cos \theta|\log|\cos \theta|\to 0 + 0 =0$$
since $|\cos \theta|\log|\cos \theta|$ is bounded.