Calculate this limit : $\lim_{x\rightarrow +\infty}\left[x\left(4\arctan\left(\frac{x+1}{x}\right)-\pi\right)\right]$ Calculate the limit
$$\lim_{x\rightarrow +\infty}\left[x\left(4\arctan\left(\frac{x+1}{x}\right)-\pi\right)\right]$$
Neither L'Hospital's rule nor Taylor expansions are allowed,

Let $4\arctan\left(1+\frac1x\right)-\pi=4y,$ then we have $x=\dfrac{1}{\tan (\pi/4+y)-1}=\dfrac{1-\tan y}{2\tan y}.$
 Now the required limit equals to $$\lim_{y\to 0}2y\left(\dfrac{1-\tan y}{\tan y}\right)=2$$ as $\lim_{y\to 0}\dfrac{\tan y}{y}=1.$
