Arithmetic Course/Non Linear Function/Exponential function/Characterizations of the exponential function

Characteristics of e

1. $\frac{d}{dt}{e}^t=e^t.$

2. $\frac{d}{dt}{\log }_{e}t=\frac{1}{t}.$

3. $e=\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n$

Similarly:

$e=\underset{x\to 0}{\lim }{(1+x)}^{1/x}$

4.$e=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots$ where n! is the factorial of n.

5. $\underset{1}{\overset{e}{\int }}\frac{1}{t}dt=1$.