Definition:Exponential Function/Real/Limit of Sequence

Definition

Let $\exp: \R \to \R_{>0}$ denote the (real) exponential function.

The exponential function can be defined as the following limit of a sequence:

$\exp x := \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$

for all $x \in \R$.

The number $\exp x$ is called the exponential of $x$.

Also see

• Real Sequence $\paren {1 + \dfrac x n}^n$ is Convergent, demonstrating that this definition is valid

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms

• Weisstein, Eric W. "Exponential Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialFunction.html