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 $a$	$$\lim_{n\rightarrow\infty}(a_n)=a$$	$$\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies |a_n-a|<\epsilon]$$ - first form $$\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies d(a_n,a)<\epsilon]$$ - Metric space $(X,d)$ $$\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}\exists U\in\mathcal{J}[a\in U\wedge(n> N \implies a_n\in U)]$$ - Topological space $(X,\mathcal{J})$
	Tending towards $+\infty$	$$\lim_{n\rightarrow\infty}(a_n)=+\infty$$	$$\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies a_n> C]$$
	Tending towards $-\infty$	$$\lim_{n\rightarrow\infty}(a_n)=-\infty$$	$$\forall C<0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies a_n< C]$$
	Diverging to $\infty$	$$\lim_{n\rightarrow\infty}(a_n)=\infty$$	$$\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies |a_n|> C]$$

(See Infinity)