Prab at 02:49, 28 October 2024

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	~~== Categories ==~~		<br>
	[[Category:Mathematical analysis]]		[[Category:Mathematical analysis]]
	[[Category:Calculus]]		[[Category:Calculus]]

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{{DISPLAYTITLE:Limit}}

'''Limit''' is a fundamental concept in [[mathematics]], particularly in the fields of [[calculus]] and [[analysis]]. It describes the behavior of a function as its argument approaches a certain point, or as the terms of a sequence approach a certain value. Limits are essential for defining derivatives, integrals, and continuity.

== Definition ==
In mathematics, the limit of a function is a fundamental concept that describes the behavior of that function as its input approaches a given point. The formal definition, known as the ''ε-δ definition of a limit'', was first introduced in the 19th century by [[Augustin-Louis Cauchy]] and [[Karl Weierstrass]].

=== Limit of a Function ===
The limit of a function ''f(x)'' as ''x'' approaches a value ''a'' is denoted as:
: \(\lim_{{x \to a}} f(x) = L\)
This notation means that as ''x'' gets closer and closer to ''a'', the function values ''f(x)'' get arbitrarily close to ''L''.

=== Limit of a Sequence ===
Similarly, the limit of a sequence \((a_n)\) is the value that the sequence approaches as the number of terms goes to infinity. It is denoted as:
: \(\lim_{{n \to \infty}} a_n = L\)
This means that for any small positive number ''ε'', there exists a natural number ''N'' such that for all ''n'' greater than ''N'', the absolute difference between ''a_n'' and ''L'' is less than ''ε''.

== Types of Limits ==
There are several types of limits in mathematical analysis:

* '''[[One-sided limit]]s''': Limits taken from only one side (left or right) of the point of interest.
* '''[[Infinite limit]]s''': Limits in which the function approaches infinity as the input approaches a certain value.
* '''[[Limit at infinity]]''': Describes the behavior of a function as the input grows without bound.

== Applications ==
Limits are used in various areas of mathematics and applied sciences:

* In [[calculus]], limits are used to define the [[derivative]] and the [[integral]].
* In [[real analysis]], limits underpin the definitions of continuity, [[metric spaces]], and [[topological spaces]].
* In [[physics]], limits describe the behavior of physical quantities under extreme conditions.

== See Also ==
* [[Derivative]]
* [[Integral]]
* [[Continuity (mathematics)]]
* [[L'Hôpital's rule]]

== Categories ==
[[Category:Mathematical analysis]]
[[Category:Calculus]]
[[Category:Mathematical concepts]]

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Prab at 02:49, 28 October 2024

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