Knowlpedia

Limit at infinity

The epsilon-M definition of the limit of a function as x goes to plus or minus infinity.

Let D⊆RD\subseteq\mathbb R be unbounded above and let f:D→Rf:D\to\mathbb R. The statement

lim⁡x→+∞f(x)=L\lim_{x\to+\infty}f(x)=L

means that for every ε>0\varepsilon>0 there exists M∈RM\in\mathbb R such that

x∈D, x>M⟹∣f(x)−L∣<ε.x\in D,\ x>M\quad\Longrightarrow\quad |f(x)-L|<\varepsilon.

If DD is unbounded below, then lim⁡x→−∞f(x)=L\lim_{x\to-\infty}f(x)=L means that for every ε>0\varepsilon>0 there exists M∈RM\in\mathbb R such that x∈Dx\in D and x<Mx<M imply ∣f(x)−L∣<ε|f(x)-L|<\varepsilon.

Examples
  • lim⁡x→∞1x=0\lim_{x\to\infty}\tfrac1x=0.
  • lim⁡x→∞2x+1x=2\lim_{x\to\infty}\tfrac{2x+1}{x}=2.
Remarks

This is the definition with closeness to a finite point replaced by movement arbitrarily far along one end of the real line.