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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> (said, "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>.
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For a [[Real numbers|real]] [[function]] <math>f(x)</math> and some value <math>c</math>, <math> \lim_{x\rightarrow c} f(x)</math> (said, "the limit of <math>f</math> at <math>x</math> as <math>x</math> goes to <math>c</math>) equals <math>L</math> [[iff]] for every <math>\epsilon > 0</math> there exists a <math>\delta </math> such that if <math>0<|x-c|<\delta</math>, then <math>|f(x)-L|< \epsilon</math>.
  
 
==Intuitive Meaning==
 
==Intuitive Meaning==
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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, <math>\displaystyle\lim_{x\rightarrow 2}x^2=4</math>, because as <math>x</math> grows arbitrarily close to 2 from either direction, the function <math>\displaystyle f(x)=x^2</math> grows arbitrarily close to 4.  In this case, the limit of the function is exactly equal to the value of the function.  That is, <math>\displaystyle \lim_{x\rightarrow c} f(x) = f(c)</math>.  This is because the function we chose was a [[continuous function]].  Unfortunately, this does not hold true in general.  For example, consider the function <math>\displaystyle f(x)</math> over the reals defined to be 0 if <math>\displaystyle x\neq 0</math> and 1 if <math>\displaystyle x=0</math>.  Although the value of the function at 0 is 1, the limit <math>\displaystyle \lim_{x\rightarrow 0}f(x)</math> is, in fact, zero.  Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, <math>\displaystyle f(x)</math> will always be close to (in fact equal to) zero.  Note that if our definition required only that <math>\displaystyle |x-c|<\delta</math>, the limit of this function would not exist.
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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, <math>\lim_{x\rightarrow 2}x^2=4</math>, because as <math>x</math> grows arbitrarily close to 2 from either direction, the function <math>f(x)=x^2</math> grows arbitrarily close to 4.  In this case, the limit of the function is exactly equal to the value of the function.  That is, <math>\lim_{x\rightarrow c} f(x) = f(c)</math>.  This is because the function we chose was a [[continuous function]].  Unfortunately, this does not hold true in general.  For example, consider the function <math>f(x)</math> over the reals defined to be 0 if <math>x\neq 0</math> and 1 if <math>x=0</math>.  Although the value of the function at 0 is 1, the limit <math>\lim_{x\rightarrow 0}f(x)</math> is, in fact, zero.  Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, <math>f(x)</math> will always be close to (in fact equal to) zero.  Note that if our definition required only that <math>|x-c|<\delta</math>, the limit of this function would not exist.
  
 
==Existence of Limits==
 
==Existence of Limits==
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Limits do not always exist.  For example <math>\displaystyle\lim_{x\rightarrow 0}\frac{1}{x}</math> does not exist, since, in fact, there exists no <math>\displaystyle \epsilon</math> for which there exists <math>\displaystyle\delta</math> satisfying the definition's conditions, since <math>\displaystyle\left|\frac{1}{x}\right|</math> grows arbitrarily large as <math>x</math> approaches 0.  However, it is possible for <math> \lim_{x\rightarrow c} f(x)</math> not to exist even when <math>f</math> is defined at <math>c</math>.  For example, consider the Dirichlet function, <math>D(x)</math>, defined to be 0 when <math>x</math> is irrational, and 1 when <math>x</math> is rational.  Here, <math>\lim_{x\rightarrow c}D(x)</math> does not exist for any value of <math>c</math>.  Alternatively, limits can exist where a function is not defined, as for the function <math>f(x)</math> defined to be 1, but only for nonzero reals.  Here, <math>\lim_{x\rightarrow 0}f(x)=1</math>, since for <math>x</math> arbitrarily close to 0, <math>f(x)=1</math>.
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Limits do not always exist.  For example <math>\lim_{x\rightarrow 0}\frac{1}{x}</math> does not exist, since, in fact, there exists no <math>\epsilon</math> for which there exists <math>\delta</math> satisfying the definition's conditions, since <math>\left|\frac{1}{x}\right|</math> grows arbitrarily large as <math>x</math> approaches 0.  However, it is possible for <math> \lim_{x\rightarrow c} f(x)</math> not to exist even when <math>f</math> is defined at <math>c</math>.  For example, consider the Dirichlet function, <math>D(x)</math>, defined to be 0 when <math>x</math> is irrational, and 1 when <math>x</math> is rational.  Here, <math>\lim_{x\rightarrow c}D(x)</math> does not exist for any value of <math>c</math>.  Alternatively, limits can exist where a function is not defined, as for the function <math>f(x)</math> defined to be 1, but only for nonzero reals.  Here, <math>\lim_{x\rightarrow 0}f(x)=1</math>, since for <math>x</math> arbitrarily close to 0, <math>f(x)=1</math>.
  
 
==Small fraction of things to be added==
 
==Small fraction of things to be added==
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{{stub}}
 
{{stub}}
  
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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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[[Category:Calculus]]
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[[Category:Definition]]

Revision as of 20:19, 15 November 2007

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

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

Existence of Limits

Limits do not always exist. For example $\lim_{x\rightarrow 0}\frac{1}{x}$ does not exist, since, in fact, there exists no $\epsilon$ for which there exists $\delta$ satisfying the definition's conditions, since $\left|\frac{1}{x}\right|$ 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


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