Difference between revisions of "Limit"

Revision as of 11:14, 28 April 2015

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 - Metric space $(X,d)$ - 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 $\infty$	$\lim_{n\rightarrow\infty}(a_n)=\infty$	$\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies \|a_n\|> C]$