Answer
0
Work Step by Step
We are given that $z=y^{2}+x^{2}$ and find the $\operatorname{limit}$:
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{e^{-1 / \sqrt{y^{2}+x^{2} }}}{\sqrt{y^{2}+x^{2} }}
\]
Given that if $(x, y) \rightarrow(0,0)$, then $z \rightarrow 0^{+}$, so we re-write the equation:
\[
\begin{array}{l}
=\lim _{z \rightarrow 0^{+}} \frac{e^{-1 / \sqrt{z}}}{\sqrt{z}} \quad \text { let } \frac{1}{\sqrt{z}}=u \\
=\lim _{u \rightarrow \infty} \frac{u}{e^{u}} \quad \text { because } z \rightarrow 0^{+} \text {So } u \rightarrow \infty \\
=0
\end{array}
\]
