Answer
$\lim\limits_{x\to\frac{\pi}{2}}\sec{x}$ does not exist.
Work Step by Step
$\lim\limits_{x\to\frac{\pi}{2}^-}\sec{x}=\lim\limits_{x\to\frac{\pi}{2}^-}\dfrac{1}{\cos{x}}=\infty.$
$\lim\limits_{x\to\frac{\pi}{2}^+}\sec{x}=\lim\limits_{x\to\frac{\pi}{2}^+}\dfrac{1}{\cos{x}}=-\infty.$
Since$\lim\limits_{x\to\frac{\pi}{2}^+}\sec{x}\ne\lim\limits_{x\to\frac{\pi}{2}^-}\sec{x}\to\lim\limits_{x\to\frac{\pi}{2}}\sec{x}$ does not exist.
