The value of _ x arrow 0 (1)/(x) [ ^ -1 ((x+1)/(2 x+1))-(π)/(4)] is...

$\lim _{x \rightarrow 0}\left(\frac{1}{x}\right)$\left[\tan ^{-1}\left(\frac{x+1}{2 x+1}\right)-\frac{\pi}{4}\right] =\lim _{x \rightarrow 0}\left(\frac{1}{x}\right)$\left[\tan ^{-1}\left(\frac{x+1}{2 x+1}\right)-\tan ^{-1}(1)\right]$ $=\lim _{x \rightarrow 0}\left(\frac{1}{x}\right) \cdot \tan ^{-1}\left(\frac{\frac{x+1}{2 x+1}-1}{1+\frac{x+1}{2 x+1}}\right)$=\lim _{x \rightarrow 0} \frac{1}{x} \cdot \tan ^{-1}\left(\frac{-x}{3 x+2}\right)$=-\lim _{x \rightarrow 0}\left[\frac{\tan ^{-1}\left(\frac{x}{3 x+2}\right)}{\frac{x}{3 x+2}} \times \frac{1}{3 x+2}\right]=-\frac{1}{2}$