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 $a$	$\lim_{n\rightarrow\infty}(a_n)=a$	$\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies \|a_n-a\|<\epsilon]$ - 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 $\infty$	$\lim_{n\rightarrow\infty}(a_n)=\infty$	$\forall C>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies \|a_n\|> C]$
Limit of a function at $x_0$	converging to $\ell$	$\lim_{x\rightarrow x_0}(f(x))=\ell$	$\forall \epsilon>0\exists\delta>0\forall x\in X\left[0<d(x,x_0)<\delta\implies d'(f(x),\ell)<\epsilon\right]$

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

@@ 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}}