I have tried to evaluate this limit:
$$\lim_{x\to \infty } \left(x(x+1) \log \left(\dfrac{x+1}{x} \right)-x\right)=\frac12$$
using $\lim_ {x\to \infty }\left(1+\dfrac{1}{x}\right)^{x}=e$, and using the variable change $z=\dfrac{1}{x}$ to get some known and standrad limit related to $\log$ natural logarithm properties function but I didn't succeed? Then any way and it's good if there is a suitable way for high school level.
Best Answer
Further to my comment, L'Hôpital's rule applied twice $$\lim\limits_{x\to\infty}\frac{(x+1)\log\left(\frac{x+1}{x}\right)-1}{\frac{1}{x}}= \lim\limits_{x\to\infty}\frac{\log\left(1 + \frac{1}{x}\right)-\frac{1}{x}}{-\frac{1}{x^2}}=\\ \lim\limits_{x\to\infty}\frac{\frac{1}{x^2 + x^3}}{\frac{2}{x^3}}= \lim\limits_{x\to\infty}\frac{1}{2}\cdot\frac{x^3}{x^2+x^3}=\frac{1}{2}$$ It is worth mentioning that both times we are dealing with $\frac{0}{0}$, so L'Hôpital's rule can be applied. L'Hôpital's rule used to be part of the high school program, I hope it still is.