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> in the [[domain]] of <math>\displaystyle f, \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 a [[Real numbers|real]] [[function]] <math>\displaystyle f(x)</math> and some value <math>\displaystyle c</math> in the [[domain]] of <math>\displaystyle f, \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= | + | 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. |
==Other Matters== | ==Other Matters== | ||
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* If a limit exists, it is unique | * If a limit exists, it is unique | ||
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{{stub}} | {{stub}} | ||
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| + | This article could use a lot of work -- there are formal errors (c doesn't have to be in the domain) 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 13:03, 30 June 2006
For a real function 
and some value 
in the domain of 
(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.
Other Matters
- Limits do not always exist
- 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 (c doesn't have to be in the domain) 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.)
