Answer
The limit does not exist.
Work Step by Step
Since
$$\lim _{x \rightarrow \frac{\pi}{2}-} \tan x=\infty$$
and
$$\lim _{x \rightarrow \frac{\pi}{2}+} \tan x=-\infty$$
Then $\lim _{x \rightarrow \frac{\pi}{2} } \tan x $ does not exist.