Answer
The limit does not exist
Work Step by Step
Substitute:
$\quad t=\sqrt{x^{2}+z^{2} +y^{2}}$
Thus:
$\frac{e^{\sqrt{x^{2}+y^{2}+z^{2}}}}{\sqrt{x^{2}+y^{2}+z^{2}}}=\frac{e^{t}}{t}$
$\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{e^{\sqrt{x^{2}+y^{2}+z^{2}}}}{\sqrt{x^{2}+y^{2}+z^{2}}}=\lim _{t \rightarrow 0+} \frac{e^{t}}{t} \quad =\frac{1}{0} =\text{does not exist}$
