TITLE: Limit of $(1+ x/n)^n$ when $n$ tends to infinity
QUESTION [44 upvotes]: Does anyone know the exact proof of this limit result?
$$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x$$

REPLY [46 votes]: $$e^{\ln{(1 + \frac{x}{n})^n} }=e^{n \ln(1+\frac{x}{n})}$$
$$\lim_{n \to +\infty} (1 + \frac{x}{n})^n
=\lim_{n \to +\infty} e^{n \ln(1+\frac{x}{n})} \\ 
=e^{\lim_{n \to +\infty} n \ln(1+\frac{x}{n})}
=e^{\lim_{n \to +\infty}\frac{ \ln(1+\frac{x}{n})}{\frac{1}{n}}}$$ 
Apply L'Hopital's Rule:
$$=e^{\lim_{n \to +\infty}\frac{(\frac{-x}{n^2})\frac{1}{1+\frac{x}{n}}}{-\frac{1}{n^2}}}
=e^{\lim_{n \to +\infty}\frac{x}{1+\frac{x}{n}}}
=e^x$$
Therefore, $$(1+\frac{x}{n})^n \to e^x$$