The Number e as a Limit

Definition

The transcendental number $e$ (Euler’s number) can be defined as the limit of $(1 + x)^{1/x}$ as $x$ approaches zero, or equivalently, as the limit of $(1 + 1/n)^n$ as $n$ approaches infinity.

• How to read: “The mathematical constant e.”
• Meaning: Euler’s number ($\approx 2.718$), base of natural logarithms—arises as the limit of continuous compound growth.

Why It Matters

The number $e$ is the “Universal Constant of Growth.” It is the base of the natural world, showing up whenever a process—be it a bank account, a virus, or a falling body—is driven by its own size. Without $e$, we couldn’t accurately model the curve of an epidemic or the discharge of a capacitor. It is the mathematical “fixed point” where a function and its own change become one, making it the most important number in calculus and the literal foundation of all continuous systems.

Core Concepts

• Limit Form (Zero): $e = \lim_{x \to 0} (1 + x)^{1/x}$

  • How to read: “The constant e is equal to the limit as x approaches zero of the quantity one plus x, raised to the power of one divided by x.”
  • Meaning: As the compounding interval shrinks to zero, growth factor approaches $e$.

• Limit Form (Infinity): $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

  • How to read: “The constant e is equal to the limit as n approaches infinity of the quantity one plus one divided by n, raised to the nth power.”
  • Meaning: Compound interest with $n$ compounding periods per year, as $n \to \infty$, yields $e$.

• Natural Logarithm Relation: $e$ is the unique base such that $\ln(e) = 1$.

  • How to read: “The natural logarithm of e is equal to one.”
  • Meaning: $e$ is the base where $\ln$ and $\exp$ are inverse functions with slope 1 at the origin.

Connected Concepts

• Precise Definition of a Limit
• Function Definition, Function Notation
• Sequence Definition and Sequence Recursion