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limsup

Represents the limit superior (supremum limit) of a sequence or function, indicating the largest possible limit value.

Overview

Essential in advanced mathematical analysis for understanding the asymptotic behavior of sequences and functions where regular limits may not exist.

  • Particularly important in real analysis and measure theory for describing sequence convergence properties
  • Often paired with liminf to establish bounds on sequence behavior
  • Commonly used in probability theory to analyze random sequences
  • Appears frequently in theoretical mathematics to prove existence of limits or characterize oscillating sequences

Examples

Upper limit of a sequence of real numbers.

lim sup⁡n→∞an=inf⁡n≥1sup⁡k≥nak\limsup_{n \to \infty} a_n = \inf_{n \geq 1} \sup_{k \geq n} a_k 
\limsup_{n \to \infty} a_n = \inf_{n \geq 1} \sup_{k \geq n} a_k

Upper limit superior in probability theory.

lim sup⁡n→∞Xn=inf⁡n≥1sup⁡k≥nXk≤M\limsup_{n \to \infty} X_n = \inf_{n \geq 1} \sup_{k \geq n} X_k \leq M 
\limsup_{n \to \infty} X_n = \inf_{n \geq 1} \sup_{k \geq n} X_k \leq M

Limit superior in analysis of function behavior.

lim sup⁡x→0sin⁡xx=1\limsup_{x \to 0} \frac{\sin x}{x} = 1 
\limsup_{x \to 0} \frac{\sin x}{x} = 1