Limit of a function: Difference between revisions

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imported>Igor Grešovnik
(added Formal definition)
imported>Igor Grešovnik
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Let  ''f''  be a function defined on an [[open interval]] containing ''a'' (except possibly at ''a'') and let ''L'' be a [[real number]].
Let  ''f''  be a function defined on an [[open interval]] containing ''a'' (except possibly at ''a'') and let ''L'' be a [[real number]].


:<math> \lim_{x \to c}f(x) = L </math>
:<math> \lim_{x \to a}f(x) = L </math>


means that
means that
:for each real &epsilon; > 0 there exists a real &delta; > 0 such that for all ''x'' with 0&nbsp;<&nbsp;|''x''&nbsp;&minus;&nbsp;''c''|&nbsp;<&nbsp;&delta;, we have  |''f''(''x'')&nbsp;&minus;&nbsp;''L''|&nbsp;<&nbsp;&epsilon;.
:for each real &epsilon; > 0 there exists a real &delta; > 0 such that for all ''x'' with 0&nbsp;<&nbsp;|''x''&nbsp;&minus;&nbsp;''a''|&nbsp;<&nbsp;&delta;, we have  |''f''(''x'')&nbsp;&minus;&nbsp;''L''|&nbsp;<&nbsp;&epsilon;.


This formal definition of function limit is due to the German mathematician [[Karl Weierstrass]].
This formal definition of function limit is due to the German mathematician [[Karl Weierstrass]].


== See also ==
== See also ==
* [[Limit (mathematics)]]
* [[Limit (mathematics)]]
*[[Limit of a sequence]]
*[[Limit of a sequence]]

Revision as of 21:23, 23 November 2007

In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either "gets close" to some point, or as it becomes arbitrarily large.

Suppose f(x) is a real-valued function and a is a real number. The expression

means that f(x) can be made arbitrarily close to L by making x sufficiently close to a. We say that "the limit of the function f of x, as x approaches a, is L".

Limit of a function can in some cases be defined even at values of the argument at which the function itself is not defined. For example,


although the function

is not defined at x=0.

Formal definition

Let f be a function defined on an open interval containing a (except possibly at a) and let L be a real number.

means that

for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − a| < δ, we have |f(x) − L| < ε.

This formal definition of function limit is due to the German mathematician Karl Weierstrass.

See also