TeXipedia

epsilon

Represents a small positive quantity or error term in mathematics and physics, commonly used in limit definitions and approximations.

Overview

A versatile Greek letter that serves multiple important roles across mathematical and scientific notation:

  • Frequently used to denote arbitrarily small positive numbers in limit proofs and epsilon-delta definitions
  • Standard symbol for error terms in statistics and numerical analysis
  • Common variable choice in physics for permittivity and efficiency calculations
  • Often appears in computer science for empty string notation and algorithm analysis

The symbol is particularly prevalent in calculus, real analysis, and engineering contexts where small quantities or margins need to be precisely specified.

Examples

Defining a small positive number in limit notation

lim⁡x→0+f(x)<ϵ\lim_{x \to 0^+} f(x) < \epsilon 
\lim_{x \to 0^+} f(x) < \epsilon

Error term in numerical analysis

∣xn−x∣<ϵ for n>N|x_n - x| < \epsilon \text{ for } n > N 
|x_n - x| < \epsilon \text{ for } n > N

Membership in set theory with small positive threshold

{x∈R:∣f(x)−L∣<ϵ}\{x \in \mathbb{R} : |f(x) - L| < \epsilon\} 
\{x \in \mathbb{R} : |f(x) - L| < \epsilon\}