# 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 real number $\delta>0$ such that, whenever $x\in X$ with $0, 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 development 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.

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

Title limit Limit 2013-03-22 12:28:25 2013-03-22 12:28:25 djao (24) djao (24) 12 djao (24) Definition msc 26A06 msc 26B12 msc 54E35 Continuous  