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Difference between revisions of "Limit"

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For a [[Real numbers|real]] [[function]] <math>\displaystyle f(x)</math> and some value <math>\displaystyle c</math>, <math> \lim_{x\rightarrow c} f(x)</math> (pronounced, "the limit of <math>f</math> of <math>x</math> as <math>x</math> goes to <math>c</math>) equals <math>\displaystyle L</math> iff for every <math>\displaystyle \epsilon > 0</math> there exists a <math>\displaystyle \delta </math> such that if <math>\displaystyle 0<|x-c|<\delta</math>, then <math>\displaystyle |f(x)-L|< \epsilon</math>.
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For a [[Real numbers|real]] [[function]] <math>\displaystyle f(x)</math> and some value <math>\displaystyle c</math>, <math> \lim_{x\rightarrow c} f(x)</math> (pronounced, "the limit of <math>f</math> at <math>x</math> as <math>x</math> goes to <math>c</math>) equals <math>\displaystyle L</math> iff for every <math>\displaystyle \epsilon > 0</math> there exists a <math>\displaystyle \delta </math> such that if <math>\displaystyle 0<|x-c|<\delta</math>, then <math>\displaystyle |f(x)-L|< \epsilon</math>.
  
 
==Intuitive Meaning==
 
==Intuitive Meaning==

Revision as of 09:57, 2 July 2006

For a real function $\displaystyle f(x)$ and some value $\displaystyle c$, $\lim_{x\rightarrow c} f(x)$ (pronounced, "the limit of $f$ at $x$ as $x$ goes to $c$) equals $\displaystyle L$ iff for every $\displaystyle \epsilon > 0$ there exists a $\displaystyle \delta$ such that if $\displaystyle 0<|x-c|<\delta$, then $\displaystyle |f(x)-L|< \epsilon$.

Intuitive Meaning

The definition of a limit is a difficult thing to grasp, so many books give an intuitive definition first: a limit is the value to which the rest of the function grows closer. For example, $\displaystyle\lim_{x\rightarrow 2}x^2=4$, because as $x$ grows arbitrarily close to 2 from either direction, the function $\displaystyle f(x)=x^2$ grows arbitrarily close to 4. In this case, the limit of the function is exactly equal to the value of the function. That is, $\displaystyle \lim_{x\rightarrow c} f(x) = f(c)$. Unfortunately, this does not hold true in general. For example, consider the function $\displaystyle f(x)$ over the reals defined to be 0 if $\displaystyle x\neq 0$ and 1 if $\displaystyle x=0$. Although the value of the function at 0 is 1, the limit $\displaystyle \lim_{x\rightarrow 0}f(x)$ is in fact zero. Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, $\displaystyle f(x)$ will always be close to (in fact equal to) zero. Note that if our definition required only that $\displaystyle |x-c|<\delta$, the limit of this function would not exist.

Existence of Limits

Limits do not always exist. For example $\displaystyle\lim_{x\rightarrow 0}\frac{1}{x}$ does not exist, since in fact there exists no $\displaystyle \epsilon$ for which there exists $\displaystyle\delta$ satisfying the definition's conditions, since $\displaystyle\left|\frac{1}{x}\right|$ grows arbitrarily large as $x$ approaches 0. However, it is possible for $\lim_{x\rightarrow c} f(x)$ not to exist even when $f$ is defined at $c$. For example, consider the Dirichlet function, $D(x)$, defined to be 0 when $x$ is irrational, and 1 when $x$ is rational. Here, $\lim_{x\rightarrow c}D(x)$ does not exist for any value of $c$. Alternatively, limits can exist where a function is not defined, as for the function $f(x)$ defined to be 1, but only for nonzero reals. Here, $\lim_{x\rightarrow 0}f(x)=1$, since for $x$ arbitrarily close to 0, $f(x)=1$.

Small fraction of things to be added

  • Limits can be added, subtracted, and mulitplied
  • If a limit exists, it is unique


This article is a stub. Help us out by expanding it.

This article could use a lot of work -- there are formal errors, and lots of omissions (limits from only one side, limits in a more general setting than the real line, continuity and its relation to limits, etc.)

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