Answer
$x=0$
Work Step by Step
We are given the function:
$f(x)=\dfrac{1}{\sqrt x\sec x}$
We should compute $\lim\limits_{x \to a} f(x)=\lim\limits_{x \to a}\dfrac{1}{\sqrt x\sec x}=\lim\limits_{x \to a}\dfrac{1}{\sqrt x\cdot\dfrac{1}{\cos x}}=\lim\limits_{x \to a}\dfrac{\cos x}{\sqrt x}$
Therefore $f(x)$ has one vertical asymptote in $x=0$.