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lim

Denotes the mathematical limit operation used to describe the behavior of functions as they approach specific values.

Overview

Essential in calculus and mathematical analysis for understanding function behavior near specific points or infinity.

  • Fundamental in defining derivatives, continuity, and convergence of sequences and series
  • Critical for evaluating asymptotic behavior in functions
  • Commonly used with subscripts to specify the approaching value
  • Appears frequently in epsilon-delta proofs and theoretical mathematics
  • Key component in studying differential equations and mathematical modeling

Examples

Finding the limit of a function as x approaches infinity

lim⁡x→∞1x=0\lim_{x \to \infty} \frac{1}{x} = 0 
\lim_{x \to \infty} \frac{1}{x} = 0

Evaluating the limit of an exponential function as x approaches 0

lim⁡x→0ex=1\lim_{x \to 0} e^x = 1 
\lim_{x \to 0} e^x = 1

Computing a one-sided limit from the left

lim⁡x→2−∣x−2∣x−2=−1\lim_{x \to 2^-} \frac{|x-2|}{x-2} = -1 
\lim_{x \to 2^-} \frac{|x-2|}{x-2} = -1