Limit $\lim_{x \to \infty}\left (1- \frac1x\right)^x = ?$ $$\lim_{x \to \infty} \left(1+ \frac1x\right)^x = e$$
, then$$\lim_{x \to \infty} \left(1- \frac1x\right)^x =\; ?$$
I tried $\lim_{x \to \infty} \left(1- \frac1x\right)^x = \lim_{x \to \infty}\left(\frac{x-1}{x}\right)^x$, but this is not helpful.

Try this:
$$
1-\frac{1}{x}=\frac{1}{\frac{x}{x-1}}=\frac{1}{1+\frac{1}{x-1}}
$$
so
$$
\lim_{x\rightarrow \infty} \left(1-\frac{1}{x}\right)^x=\lim_{x\rightarrow \infty} \left(\frac{1}{1+\frac{1}{x-1}}\right)^x=\lim_{x\rightarrow \infty} \frac{1}{\left(1+\frac{1}{x-1}\right)^{x-1}} \left(\frac{1}{1+\frac{1}{x-1}} \right)=\frac{1}{e}\cdot 1=\frac{1}{e}
$$
