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,

A solution without L'Hospital and Taylor which uses only the geometric series and basic differential calculus
Consider the function
$$
f(x)=4\arctan\left(\frac{x+1}{x}\right),\quad f(\infty)=4\arctan(1)=\pi
$$
Now
$$
f'(x)=-\frac{4}{2 x (x+1)+1}
$$
By a simple application of the geometric series we see that
$$
f'(x)\sim_{\infty}-\frac{2}{x^2}
$$
or
$$
f(x)\sim_{\infty}\frac{2}{x}+C
$$
where $C$ is fixed by the conditon $f(\infty)=\pi$ so
$$
f(x)\sim_{\infty}\frac{2}{x}+\pi
$$
so we have

$$
\lim_{x\rightarrow\infty}[x(f(x)-\pi)]=2
$$

