Answer
The limit does not exist.
Work Step by Step
We have
$$
\lim _{\theta \rightarrow \frac{\pi}{2}} \theta \sec\theta=\frac{\pi/2}{0}.
$$
$$
\lim _{\theta \rightarrow (\frac{\pi}{2})^+} \theta \sec\theta=\frac{\pi/2}{0^-}=-\infty, $$
$$
\lim _{\theta \rightarrow (\frac{\pi}{2})^-} \theta \sec\theta=\frac{\pi/2}{0^+}=\infty, $$
so the limit does not exist.
