Evaluate `lim_(x to 0) x[tan^(-1)((x+1)/(x+2))-tan^(-1)((x)/(x+2))].`

Explanation

To evaluate the limit

\[ \lim_{x \to 0} x \left( \tan^{-1} \left( \frac{x+1}{x+2} \right) - \tan^{-1} \left( \frac{x}{x+2} \right) \right), \]

we can use the identity for the difference of inverse tangents:

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