# 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] $\lim_{n\rightarrow\infty}(a_n)=a$
• $\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies |a_n-a|<\epsilon]$ - 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 [ilmath](X,d)[/ilmath]
• $\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}\exists U\in\mathcal{J}[a\in U\wedge(n> N \implies a_n\in U)]$ - Topological space [ilmath](X,\mathcal{J})[/ilmath]
Tending towards [ilmath]+\infty[/ilmath] $\lim_{n\rightarrow\infty}(a_n)=+\infty$
• $\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies a_n> C]$
Tending towards [ilmath]-\infty[/ilmath] $\lim_{n\rightarrow\infty}(a_n)=-\infty$
• $\forall C<0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies a_n< C]$
Diverging to [ilmath]\infty[/ilmath] $\lim_{n\rightarrow\infty}(a_n)=\infty$
• $\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies |a_n|> C]$
Limit of a function at [ilmath]x_0[/ilmath] converging to [ilmath]\ell[/ilmath] $\lim_{x\rightarrow x_0}(f(x))=\ell$ $\forall \epsilon>0\exists\delta>0\forall x\in X\left[0<d(x,x_0)<\delta\implies d'(f(x),\ell)<\epsilon\right]$

TODO: I like the idea of a summary page, but it needs to link to the right pages and have definitions in place

(See Infinity)