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).

Limiting tan⁡−1\tan^{-1}tan−1 at infinity