Difference between revisions of "Limit (mathematics)"
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| − | Let <math>f(x)</math> be a real valued function in one variable. We say that | + | Let <math>f(x)</math> be a real valued [[function]] in one [[variable]]. We say that |
<math> | <math> | ||
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| − | if for any error <math>\varepsilon</math>, we can find a sufficiently small | + | if for any error <math>\varepsilon</math>, we can find a sufficiently small neighborhood of <math>p</math> so that <math>f(x)</math> is within <math>\varepsilon</math> of the value <math>L</math> for any <math>x\neq p</math> in that neighborhood. One says that the limit of ''f(x)'' at ''p'' exists and is equal to ''L''. |
The standard, though more verbose, way of saying this is that for any <math>\varepsilon > 0</math> there exists a sufficiently small <math>\delta > 0</math> such that <math>|f(x)-L| <\varepsilon</math> whenever <math>0<|x-p|<\delta</math>. | The standard, though more verbose, way of saying this is that for any <math>\varepsilon > 0</math> there exists a sufficiently small <math>\delta > 0</math> such that <math>|f(x)-L| <\varepsilon</math> whenever <math>0<|x-p|<\delta</math>. | ||
[[category:mathematics]] | [[category:mathematics]] | ||
Revision as of 17:44, August 1, 2008
Let be a real valued function in one variable. We say that
if for any error , we can find a sufficiently small neighborhood of so that is within of the value for any in that neighborhood. One says that the limit of f(x) at p exists and is equal to L.
The standard, though more verbose, way of saying this is that for any there exists a sufficiently small such that whenever .
