limit

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

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

if, for every real number $\\varepsilon>0$ , there exists a realnumber $\\delta>0$ 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–deservedreputation for being notoriously hard for inexperienced students tomaster. There is no easy fix for this problem, since the concept of alimit is inherently difficult to state precisely (and indeed wasn’teven accomplished historically until the 1800’s by Cauchy, well afterthe development of calculus in the 1600’s by Newton and Leibniz).However, there are number of related definitions, which, takentogether, may shed some light on the nature of the concept.

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The notion of a limit can be generalized to mappings between arbitrarytopological spaces, under some mild restrictions. In this context we say that $\\lim_{x\\to a}f(x)=L$ if $a$ is a limit point of $X$ and, for everyneighborhood $V$ of $L$ (in $Y$ ), there is a deleted neighborhood $U$ of $a$ (in $X$ ) which is mapped into $V$ by $f$ . One also requires that the range $Y$ be Hausdorff (or at least $T_{1}$ ) in order to ensure that limits, when they exist, are unique.
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Let $a_{n},n\\in\\mathbb{N}$ be a sequence of elements in a metricspace $X$ . We say that $L\\in X$ is the limit of the sequence, iffor every $\\varepsilon>0$ there exists a natural number $N$ suchthat $d(a_{n},L)<\\varepsilon$ for all natural numbers $n>N$ .
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The definition of the limit of a mapping can be based on thelimit of a sequence. To wit, $\\lim_{x\\to a}f(x)=L$ if and onlyif, for every sequence of points $x_{n}$ in $X$ converging to $a$ (that is, $x_{n}\\to a$ , $x_{n}\eq a$ ), the sequence of points $f(x_{n})$ in $Y$ converges to $L$ .

In calculus, $X$ and $Y$ are frequently taken to be Euclidean spaces $\\mathbb{R}^{n}$ and $\\mathbb{R}^{m}$ , in which case the distancefunctions $d_{X}$ and $d_{Y}$ cited above are just Euclidean distance.