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. $e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

Similarly:

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

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

5. $\int _{1}^{e}{\frac {1}{t}}\,dt={1}$.