Answer
$-\displaystyle \frac{\pi}{2}$
Work Step by Step
$y=\tan^{-1}x$ is the number in $(-\pi/2, \pi/2)$ for which $\tan y=x.$
In $(-\pi/2, \pi/2)$, we want the angle (in radians) for which $\tan x\rightarrow+\infty$.
This is $-\displaystyle \frac{\pi}{2}$.
Alternatively, we can reach the same conclusion by observing the graph of $y=\tan^{-1}x$ (also written as $\arctan x$) when $ x\rightarrow-\infty$. See below.
