Calculate the limit $\lim_{(x,y,z)->(0,0,0)}{\frac{xyz}{\sqrt{x} + \sqrt{y} + \sqrt{z}}}$ $$\lim_{(x,y,z)->(0,0,0)}{\frac{xyz}{\sqrt{x} + \sqrt{y} + \sqrt{z}}}$$
I tried thinking about a solution using the epsilon-delta definition: I assumed that the limit is 0 but I couldn't figure out a inequality that would lead me to the solution.

Using the AM-GM inequality
$$0 \le \frac{xyz}{\sqrt{x}+\sqrt{y}+\sqrt{z}} = \frac{\left( \sqrt[3]{\sqrt{x}\sqrt{y}\sqrt{z}} \right)^{3/2}}{\sqrt{x}+\sqrt{y}+\sqrt{z}} \le \frac{\left( \frac{\sqrt{x}+\sqrt{y}+\sqrt{z}}{3} \right)^{3/2}}{\sqrt{x}+\sqrt{y}+\sqrt{z}} = \frac{1}{3\sqrt{3}}\sqrt{\sqrt{x}+\sqrt{y}+\sqrt{z}}$$
so by the squeeze theorem the limit is $0$.
