PlanetMath: limit

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limit

(Definition)

Let and be metric spaces and let $a \in X$ be a limit point of . Suppose that $f\colon X \setminus \{a\} \to Y$ is a function defined everywhere except at . For $L \in Y$ , we say the limit of as approaches is equal to , or

$\displaystyle \lim_{x \to a} f(x) = L$

if, for every real number $\varepsilon > 0$ , there exists a real number $\delta > 0$ such that, whenever $x \in X$ with $0 < d_X(x,a) < \delta$ , then $d_Y(f(x), L) < \varepsilon$ .

The formal definition of limit as given above has a well-deserved reputation for being notoriously hard for inexperienced students to master. There is no easy fix for this problem, since the concept of a limit is inherently difficult to state precisely (and indeed wasn't even accomplished historically until the 1800's by Cauchy, well after the invention of calculus in the 1600's by Newton and Leibniz). However, there are number of related definitions, which, taken together, may shed some light on the nature of the concept.

The notion of a limit can be generalized to mappings between arbitrary topological spaces, under some mild restrictions. In this context we say that $\lim_{x \to a} f(x) = L$ if is a limit point of and, for every neighborhood of (in ), there is a deleted neighborhood of (in ) which is mapped into by . One also requires that the range be Hausdorff (or at least ) in order to ensure that limits, when they exist, are unique.
Let $a_n, n\in\mathbb{N}$ be a sequence of elements in a metric space . We say that $L\in X$ is the limit of the sequence, if for every $\varepsilon>0$ there exists a natural number such that $d(a_n,L)< \varepsilon$ for all natural numbers .
The definition of the limit of a mapping can be based on the limit of a sequence. To wit, $\lim_{x \to a} f(x) = L$ if and only if, for every sequence of points in converging to (that is, $x_n \to a$ , $x_n \neq a$ ), the sequence of points in converges to .

In calculus,

and

are frequently taken to be Euclidean spaces $\mathbb{R}^n$ and $\mathbb{R}^m$ , in which case the distance functions

and

cited above are just Euclidean distance.

"limit" is owned by djao. [ full author list (2) | owner history (1) ]

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See Also: continuous

Attachments:: one-sided limit (Definition) by CWoo
improper limits (Definition) by pahio
limit rules of functions (Theorem) by pahio
iterated limit in $\mathbb{R}^2$ (Definition) by pahio
Cauchy condition for limit of function (Theorem) by pahio

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Cross-references: Euclidean distance, distance, Euclidean spaces, converges, points, natural number, sequence, order, Hausdorff, range, deleted neighborhood, neighborhood, restrictions, topological spaces, mappings, definitions, Calculus, even, fix, real number, function, limit point, metric spaces
There are 178 references to this entry.

This is version 8 of limit, born on 2002-02-25, modified 2006-07-12.
Object id is 2662, canonical name is Limit.
Accessed 14472 times total.

Classification:

AMS MSC:	26A06 (Real functions :: Functions of one variable :: One-variable calculus)
	26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)
	54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)

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