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 $e^x$,
$$\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}=
\lim_{n \to \infty}e^{\log[(1+\frac{1}{n})^n-(1+\frac{1}{n})](-n)}=e^{-\infty}=0
$$