Trigonometric Limit!

Calculus Level 2

Evaluate

$\displaystyle \lim_{x \to 999^+} \tan^{-1} \left(\frac{1}{x - 999}\right).$

Details and assumptions

$\tan^{-1}x$ denotes the inverse of $\tan x$ and not the reciprocal $\frac{1}{\tan x}$ .

The principal branch of $\tan^{-1}$ is $\left( -\frac{\pi}{2}, \frac{\pi}{2}\right)$ .