How do I calculate
$$\lim_{n\to\infty} n-ne\left(1-\frac{1}{n}\right)^n$$
Edit: changed it to correct question with $1-1/n$
2026-05-15 03:49:51.1778816991
Limit of $\lim_{n\to\infty} n-ne\left(1-\frac{1}{n}\right)^n$?
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\begin{align} n-ne\left(1-\frac{1}{n}\right)^n &= n-nee^{n\ln\left(1-\frac{1}{n}\right)} \\ &= n-nee^{\left(-1-\frac{1}{2n}+O(\frac{1}{n^2})\right)} \\ &= n-n\left(1-\frac{1}{2n}+O(\frac{1}{n^2})\right) \\ &= \frac{1}{2}+O(\frac{1}{n}) \end{align} then $$\lim_{n\to\infty} n-ne\left(1+\frac{1}{n}\right)^n=\color{blue}{\frac12}$$