Difference between revisions of "Limit"
| Line 1: | Line 1: | ||
| − | For a [[Real numbers|real]] [[function]] <math>\displaystyle f(x)</math> and some value <math>\displaystyle c</math> | + | 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>. |
==Intuitive Meaning== | ==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, <math>\displaystyle\lim_{x\rightarrow 2}x^2=4</math>, because as the function <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>. 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. | 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 the function <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>. 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. | ||
| − | == | + | ==Existence of Limits== |
| − | + | 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>. | |
| + | |||
| + | ==Small fraction of things to be added== | ||
* Limits can be added, subtracted, and mulitplied | * Limits can be added, subtracted, and mulitplied | ||
* If a limit exists, it is unique | * If a limit exists, it is unique | ||
| Line 12: | Line 14: | ||
{{stub}} | {{stub}} | ||
| − | This article could use a lot of work -- there are formal errors | + | 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.) |
Revision as of 20:30, 30 June 2006
For a real function 
and some value 
, 
(pronounced, "the limit of 
of 
as goes to 
) equals 
iff for every 
there exists a 
such that if 
, then 
.
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, 
, because as the function grows arbitrarily close to 2 from either direction, the function 
grows arbitrarily close to 4. In this case, the limit of the function is exactly equal to the value of the function. That is, 
. Unfortunately, this does not hold true in general. For example, consider the function over the reals defined to be 0 if 
and 1 if 
. Although the value of the function at 0 is 1, the limit 
is in fact zero. Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, will always be close to (in fact equal to) zero. Note that if our definition required only that 
, the limit of this function would not exist.
Existence of Limits
Limits do not always exist. For example 
does not exist, since in fact there exists no 
for which there exists 
satisfying the definition's conditions, since 
grows arbitrarily large as approaches 0. However, it is possible for not to exist even when is defined at . For example, consider the Dirichlet function, 
, defined to be 0 when is irrational, and 1 when is rational. Here, 
does not exist for any value of . Alternatively, limits can exist where a function is not defined, as for the function 
defined to be 1, but only for nonzero reals. Here, 
, since for arbitrarily close to 0, 
.
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.)
