<s>[INST] $\lim_{(x,y,z) \to (0,0,0)} \frac{xyz}{x^2+y^2+z^2}=0$ 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.
 [/INST] Note that $|x|\le \sqrt{x^2+y^2+z^2}$, $|y|\le \sqrt{x^2+y^2+z^2}$, and $|z|\le \sqrt{x^2+y^2+z^2}$.
Hence, given $\varepsilon>0$,
$$ \left| \frac{xyz}{x^2+y^2+z^2}\right|\le \sqrt{x^2+y^2+z^2}<\varepsilon$$
whenever $\sqrt{x^2+y^2+x^2}<\delta=\varepsilon$.
And we are done!
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