another possible definition of e? many writings about the fame of e.

\displaystyle\frac{d}{dx}e^x = \frac{d}{dx}\lim_{n\to\infty}(1+\frac{x}{n})^n

= \displaystyle\lim_{n\to\infty}(\frac{d}{dx}(1+\frac{x}{n})^n) (uniform convergence)

= \displaystyle\lim_{n\to\infty}n(1+\frac{x}{n})^{n-1}.\frac{1}{n}

= \displaystyle\frac{lim_{n\to\infty}(1+\frac{x}{n})^n}{lim_{n\to\infty}(1+\frac{x}{n})} (assume both limits exist)

= \displaystyle\frac{e^x}{1}.

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