This is a question that follows doubts that showed up in another question. I have tried to use Taylor expansions to prove the following unsuccessfully:
$\lim_{n \to \infty}\left(1 + \frac{x}{n}\right)^n = e^x$
in order to prove:$e^{-x}=\lim_{n \to \infty}\left(1 - \frac{x}{n}\right)^n$
Could someone prove it?
Thanks in advance!
Let $$s = \lim_{n\to \infty} \left(1+\frac{x}{n} \right)^n$$ then $$\ln(s) = \lim_{n\to \infty} n\ln\left(1+\frac{x}{n}\right) = \lim_{n\to \infty} \frac{\ln\left(1+\frac{x}{n}\right)}{1/n}$$ Using l'Hopital's rule, $$= \lim_{n\to \infty} \frac{\left(1+\frac{x}{n}\right)^{-1}(-x/n^2)}{-1/n^2} = \lim_{n\to\infty} \frac{x}{1+\frac{x}{n}} = x$$ and so $s = e^x$.
