Give me hints for evaluating this limit Evaluate this limit:
$$\lim_{n\to\infty}{\left\{\left(1+\frac{1}{n}\right)^n-\left(1+\frac{1}{n}\right)\right\}}^{-n}$$
Please give me some hints. If you provide a complete answer instead, please include a spoiler tag. 

HINT:
$$\lim_{n\to\infty}\left(\left(1+\frac{1}{n}\right)^n-\left(1+\frac{1}{n}\right)\right)^{-n}=$$
$$\lim_{n\to\infty}\left(-1+\left(1+\frac{1}{n}\right)^n-\frac{1}{n}\right)^{-n}=$$
$$\lim_{n\to\infty}\exp\left(\ln\left(\left(-1+\left(1+\frac{1}{n}\right)^n-\frac{1}{n}\right)^{-n}\right)\right)=$$
$$\lim_{n\to\infty}\exp\left(-n\ln\left(-1+\left(1+\frac{1}{n}\right)^n-\frac{1}{n}\right)\right)=$$
$$\exp\left(\lim_{n\to\infty}-n\ln\left(-1+\left(1+\frac{1}{n}\right)^n-\frac{1}{n}\right)\right)=$$
$$\exp\left(-\lim_{n\to\infty}n\lim_{n\to\infty}\ln\left(-1+\left(1+\frac{1}{n}\right)^n-\frac{1}{n}\right)\right)$$
