Difference between revisions of "Limit"

Latest revision as of 18:23, 8 January 2016

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]
Limit of a function at [ilmath]x_0[/ilmath]	converging to [ilmath]\ell[/ilmath]	[math]\lim_{x\rightarrow x_0}(f(x))=\ell[/math]	[math]\forall \epsilon>0\exists\delta>0\forall x\in X\left[0<d(x,x_0)<\delta\implies d'(f(x),\ell)<\epsilon\right][/math]

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

@@ Line 32: / Line 32: @@
 |
 * <math>\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies |a_n|> C]</math>
+|-
+| Limit of a function at {{M|x_0}}
+| converging to {{M|\ell}}
+| {{MM|1=\lim_{x\rightarrow x_0}(f(x))=\ell}}
+| {{MM|1=\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]])
 {{Definition|Real Analysis}}