<s>[INST] Evaluate $\lim_{(x,y,z)\to (0,0,0)}\frac{1}{x^2+y^2+z^2}e^{-\frac{1}{\sqrt{x^2+y^2+z^2}}}$ 
Evaluate $$\lim_{(x,y,z)\to (0,0,0)}\frac{1}{x^2+y^2+z^2}e^{-\frac{1}{\sqrt{x^2+y^2+z^2}}}$$

Let $t=\sqrt{x^2+y^2+z^2}$
$$\lim_{t\to 0}\frac{1}{t^2}e^{-\frac{1}{t}}$$
Let $r=\frac{1}{t}$
$$\lim_{r\to \infty}r^2 e^{-r}=\lim_{r\to \infty}\frac{r^2}{e^r}$$
using l'hopital
$$\lim_{r\to \infty}\frac{2r}{e^r}$$
using l'hopital again
$$\lim_{r\to \infty}\frac{2}{e^r}=0$$
Is the following valid? is there is a simpler way?
 [/INST] HINT
By spherical coordinates $\sqrt{x^2+y^2+z^2}=r\to 0$
$$\frac{1}{ x^2+y^2+z^2 }e^{-\frac{1}{\sqrt{x^2+y^2+z^2}}}=\frac1{r^2}e^{-\frac1r}$$
and for $y=\frac1r\to \infty$ since eventually $e^y\ge y^3$
$$\frac1{r^2}e^{-\frac1r}=\frac{y^2}{e^y}\le\frac{y^2}{y^3}=\frac1y\to 0$$
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