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Limit of a function at infinity

The value L that f(x) approaches as x→∞, defined by an ε–M condition.
Limit of a function at infinity

Let f:E→Rf:E\to\mathbb{R} (or C\mathbb{C}) where E⊆RE\subseteq\mathbb{R} is unbounded above. We say

lim⁡x→∞f(x)=L\lim_{x\to\infty} f(x)=L

if

∀ε>0, ∃M∈R such that ∀x∈E, (x>M⇒∣f(x)−L∣<ε).\forall \varepsilon>0,\ \exists M\in\mathbb{R}\ \text{such that}\ \forall x\in E,\ \bigl(x>M \Rightarrow |f(x)-L|<\varepsilon\bigr).

Limits at infinity formalize the long-range behavior of functions and are used in asymptotics, improper integrals, and series tests.

Examples:

  • lim⁡x→∞1x=0\lim_{x\to\infty} \frac{1}{x}=0.
  • lim⁡x→∞xx+1=1\lim_{x\to\infty} \frac{x}{x+1}=1.
  • lim⁡x→∞sin⁡x\lim_{x\to\infty} \sin x does not exist (oscillation).