Find the limit (if it exists): $\lim_{(x,y,x)\rightarrow(0,0,0)}\frac{xyz}{x^2+y^2+z^2}$ Find the limit (if it even exists). If not, prove it doesn't exist.
        $$\lim_{(x,y,x)\rightarrow(0,0,0)}\frac{xyz}{x^2+y^2+z^2}$$

If you use the facts 
$$ |x|=\sqrt{x^2}\leq \sqrt{x^2+y^2+z^2}, \quad |y|=\sqrt{y^2}\leq \sqrt{x^2+y^2+z^2},\quad |z|=\sqrt{z^2}\leq \sqrt{x^2+y^2+z^2}, $$
then the problem falls apart as
$$ \Bigg|\frac{xyz}{x^2+y^2+z^2}\Bigg|= \frac{|xyz|}{x^2+y^2+z^2}= \frac{|x||y||z|}{x^2+y^2+z^2} $$
$$ \leq \frac{(x^2+y^2+z^2)^{3/2}}{x^2+y^2+z^2}=\sqrt{x^2+y^2+z^2}< \epsilon=\delta. $$
