Answer
$x=0$
Work Step by Step
We are given the function:
$g(x)=e^{1/x}$
We should compute $\lim\limits_{x \to a} g(x)=\lim\limits_{x \to a}e^{1/x}$
The only value of $x$ for which the function is undefined is $x=0$.
$\lim\limits_{x \to 0^-} g(x)=\lim\limits_{x \to 0^-}e^{1/x}=0$
$\lim\limits_{x \to 0^+} g(x)=\lim\limits_{x \to 0^+}e^{1/x}=\infty$
Therefore $g(x)$ has one vertical asymptote in $x=0$.