Saturday, April 8, 2017

How to find limit limlimitsxrightarrow+inftytanleft(fracpix2x+1right)1/x




I need to find this limit lim.



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}


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