Limit

Definition

A limit allows us to sidestep the notion of infinity and to allow us to potentially extend the domain of functions

Class	Name	Form	Meaning
Limit of a sequence	converging to [ilmath]a[/ilmath]	[math]\lim_{n\rightarrow\infty}(a_n)=a[/math]	[math]\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies |a_n-a|<\epsilon] [/math] - first form [math]\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies d(a_n,a)<\epsilon] [/math] - Metric space [ilmath](X,d)[/ilmath] [math]\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}\exists U\in\mathcal{J}[a\in U\wedge(n> N \implies a_n\in U)][/math] - Topological space [ilmath](X,\mathcal{J})[/ilmath]
	Tending towards [ilmath]+\infty[/ilmath]	[math]\lim_{n\rightarrow\infty}(a_n)=+\infty[/math]	[math]\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies a_n> C][/math]
	Tending towards [ilmath]-\infty[/ilmath]	[math]\lim_{n\rightarrow\infty}(a_n)=-\infty[/math]	[math]\forall C<0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies a_n< C][/math]
	Diverging to [ilmath]\infty[/ilmath]	[math]\lim_{n\rightarrow\infty}(a_n)=\infty[/math]	[math]\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies |a_n|> C][/math]

(See Infinity)