Exp x equals e^x

Theorem

 * $\exp x = e^x$

where:


 * $\exp$ is the exponential function


 * $e$ is Euler's number


 * $e^x$ is $e$ to the $x$th power

Proof
Let the restriction of the exponential function to the rationals be defined as:


 * $\displaystyle \exp \restriction_\Q: x \mapsto \lim_{n \mathop \to +\infty}\left ({1 + \frac x n}\right)^n$

Let $e$ be Euler's Number defined as:


 * $e = \displaystyle \lim_{n \mathop \to +\infty}\left ({1 + \frac 1 n}\right)^n$

For $x=0$:

For $x \ne 0$:

For $x \in \R \setminus \Q$, we invoke Power Function to Rational Power permits Unique Continuous Extension.