What you have done is wrong because you assumed the limit existed.
We must evaluate $\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}= \lim_{n \to \infty}e^{\log[(1+\frac{1}{n})^n-(1+\frac{1}{n})](-n)} $ Remember $\lim_{n \to \infty}(1+\frac{1}{n})^n=e$ Thus, because $e>2$ $\lim_{n \to \infty}\log[(1+\frac{1}{n})^n-(1+\frac{1}{n})](-n)=\log(e-1)(-\infty)=-\infty $ By continuity of $e^x$, $\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}= \lim_{n \to \infty}e^{\log[(1+\frac{1}{n})^n-(1+\frac{1}{n})](-n)}=e^{-\infty}=0 $