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

Characteristics of e

 * [[Image:hyperbola E.svg|200px]]

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