How to evaluate this limit without using L'Hôpital's rule: $\lim\limits_{n \to \infty}\left\{n\left[\ln{\left(n+1\right)}-\ln{n}\right]\right\}$?
Find $\lim _{x\to \infty }\left(x\left(\ln\left(x+1\right)-\ln x\right)\right) $