Difference between revisions of "Limit (mathematics)"
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if for any error <math>\varepsilon</math>, we can find a sufficiently small neighbhorhood 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''. | if for any error <math>\varepsilon</math>, we can find a sufficiently small neighbhorhood 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|< | + | 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 20:18, July 4, 2008
Let be a real valued function in one variable. We say that
if for any error , we can find a sufficiently small neighbhorhood 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 .
