TITLE: Show $\lim\limits_{n\to\infty} n \ln\left(1-\frac{1}n\right) = -1$
QUESTION [7 upvotes]: Could you help me show that $$\lim\limits_{n\to\infty} n \ln \left({1-\frac{1}n} \right) = -1 ?$$

REPLY [5 votes]: $$\lim\limits_{n\to\infty} n \ln\left({1-\frac{1}n}\right)=\lim_{n\to\infty}  \ln\left({1-\frac{1}n}\right)^n=\ln\lim_{n\to\infty}\left(1-{1\over n}\right)^n=\ln e^{-1}=-1$$