Limiting \(\tan^{-1}\) at infinity

Calculus Level 4

\[\large \lim_{x\rightarrow \infty}x \left[ \tan^{-1} \left ( \frac{x+1}{x+2} \right )-\tan^{-1}\left ( \frac{x}{x+2} \right) \right] = \, ? \]

\[\]Enter your answer as 999 if you think the limit does not exist (or it diverges).