$\lim_{(x,y,z) \to (0,0,0)} \frac{xyz}{x^2+y^2+z^2}=0$ How to show that
$$\lim_{(x,y,z) \to (0,0,0)} \frac{xyz}{x^2+y^2+z^2}=0,$$ where $x,y,z>0$.
My attempt:
$$||(x,y,z)|| < \delta \implies |x|, |y|, |z| < \delta$$
$$\left | \frac{xyz}{x^2+y^2+z^2} \right | < \left | \frac{xyz}{x^2}\right | < \frac{\delta^3}{x^2}.$$
Now, I do not know how to proceed, and I think my attempt might be wrong.

Using A-G inequality, $x^2+y^2+z^2\geq3(xyz)^{\frac{2}{3}}$
