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

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

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