How to find limit $lim limits_{ xrightarrow +infty }{ tan { { left( frac { pi x }{ 2x+1 } right) }^{ 1/x } } } $

I need to find this limit $\lim\limits_{ x\rightarrow +\infty }{ \tan { { \left( \frac { \pi x }{ 2x+1 } \right) }^{ 1/x } } } $.

Give a hint please.Thanks

Answer

Hints:
$$\frac{\pi x}{2x+1} = \frac{\pi}{2}-\frac{\pi}{4x+2}\tag{1}$$
$$\tan\left(\frac{\pi x}{2x+1}\right) = \cot\left(\frac{\pi}{4x+2}\right)\tag{2}$$
$$\frac{1}{x}\,\log\cot\left(\frac{\pi}{4x+2}\right)=\frac{\log\frac{4}{\pi}+\log x}{x}+O\left(\frac{1}{x^2}\right)\text{ as }x\to+\infty.\tag{3}$$