Answer
$\displaystyle \frac{\pi}{2}$
Work Step by Step
$y=\sin^{-1}x$ is the number in $[-\pi/2, \pi/2]$ for which $\sin y=x.$
As the value of sine approaches 1 (from the left, because it can't approach it from the right - sine is never greater than 1), the angle we approach (in radians) is $\displaystyle \frac{\pi}{2}$.
Alternatively, we can reach the same conclusion by observing the graph of $y=\sin^{-1}x$ (also written as $\arcsin x$) in the vicinity of x=1. See below.
