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

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Characteristics of e[edit]

Hyperbola E.svg

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}.