Answer
$+\infty$
Work Step by Step
$\displaystyle \sec x=\frac{1}{\cos x}$
You can sketch a unit circle. 0 radians is at (1,0)$;$ the positive direction is counterclockwise; the negative direction is clockwise$;$
$-\displaystyle \frac{\pi}{2}$ radians is positioned at $(0,-1).$
When $x\displaystyle \rightarrow(-\frac{\pi}{2})^{+}$, x is a radian angle measure in the fourth quadrant, (slightly more than $-\displaystyle \frac{\pi}{2}$ means on the counterclockwise side of $-\displaystyle \frac{\pi}{2}$) where cosine is positive.
The numerator, $\rightarrow 1$, (positive) and the denominator, $\cos x\rightarrow 0$, (positive).
Thus, $\displaystyle \sec x=\frac{1}{\cos x}\rightarrow+\infty$
