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Limit (mathematics)

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In mathematics, the concept of a "'''limit'''" is used to describe the behavior of a function (mathematics) function as its argument either gets "close" to some point, or as it becomes larger and larger; or the behavior of a sequence's elements, as their index becomes larger and larger. Limits are used in calculus and other branches of mathematical analysis to define derivatives and Continuous function continuity. The concept of the "limit of a function" is further generalized to the concept of net (topology) topological net, while the limit of a sequence is closely related to limit (category theory) limit and direct limit in category theory. Mathematics students usually first encounter limits in introductory calculus classes, and understanding the detailed concept often presents a stumbling block. Readers seeking an introductory explanation might look at the Wikibooks Calculus section about limits [http://en.wikibooks.org/wiki/Calculus:Limits]. This present article does have some elementary exposition but it's also about how limits are treated in more advanced branches of mathematics.

Limit of a function
''Main article: limit of a function'' Suppose ''f''(''x'') is a real function and ''c'' is a real number. The expression: :$\backslash lim\_\{x\; \backslash to\; c\}f(x)\; =\; L$ means that ''f''(''x'') can be made to be as close to ''L'' as desired by making ''x'' sufficiently close to ''c''. In that case, we say that "the limit of ''f''(''x''), as ''x'' approaches ''c'', is ''L''". Note that this statement can be true even if ''f''(''c'') $\backslash neq$ ''L''. Indeed, the function ''f''(''x'') need not even be defined at ''c''. Two examples help illustrate this. Consider $f(x)\; =\; \backslash frac\{x\}\{x^2\; +\; 1\}$ as ''x'' approaches 2. In this case, ''f''(''x'') is defined at 2 and equals its limit of 0.4: {|border="1" cellspacing="0" cellpadding="4" |f(1.9)||f(1.99)||f(1.999)||f(2)||f(2.001)||f(2.01)||f(2.1) |- |0.4121||0.4012||0.4001||$\backslash Rightarrow$ 0.4 $\backslash Leftarrow$||0.3998||0.3988||0.3882 |} As ''x'' approaches 2, ''f''(''x'') approaches 0.4 and hence we have $\backslash lim\_\{x\backslash to\; 2\}f(x)=0.4$. In the case where $f(c)\; =\; \backslash lim\_\{x\backslash to\; c\}\; f(x)$, ''f'' is said to be continuous function continuous at ''x'' = ''c''. But it is not always the case. Consider : The limit of ''g''(''x'') as ''x'' approaches 2 is 0.4 (just as in ''f''(''x'')), but $\backslash lim\_\{x\backslash to\; 2\}g(x)\backslash neq\; g(2)$; ''g'' is not continuous at ''x'' = 2. Or, consider the case where ''f''(''x'') is undefined at ''x'' = ''c''. :$f(x)\; =\; \backslash frac\{x\; -\; 1\}\{\backslash sqrt\{x\}\; -\; 1\}$ In this case, as ''x'' approaches 1, ''f''(''x'') is undefined at ''x'' = ''1'' but the limit equals 2: {|border="1" cellspacing="0" cellpadding="4" |f(0.9)||f(0.99)||f(0.999)||f(1.0)||f(1.001)||f(1.01)||f(1.1) |- |1.95||1.99||1.999||$\backslash Rightarrow$ undef $\backslash Leftarrow$||2.001||2.010||2.10 |} Thus, ''x'' can get as close to ''1'', so long as it is not equal to ''1'', so that the limit of $f(x)$ is ''2''.

Formal definition
A limit is formally defined as follows: Let $f$ be a function defined on an open interval containing $c$ (except possibly at $c$) and let $L$ be a real number. The statement :$\backslash lim\_\{x\; \backslash to\; c\}f(x)\; =\; L$ means that for each $\backslash varepsilon\backslash \; >0$ there exists a $\backslash delta\backslash \; >0$ such that for all $x$ where , then .

Limit of a function at infinity
A related concept to limits as ''x'' approaches some finite number is the limit as ''x'' approaches positive or negative infinity. This does not literally mean that the difference between ''x'' and infinity becomes small, since infinity is not a number; rather, it means that ''x'' either becomes larger and larger (for positive infinity) or smaller and smaller (for negative infinity). For example, consider $f(x)\; =\; \backslash frac\{2x\}\{x\; +\; 1\}$. * ''f''(100) = 1.9802 * ''f''(1000) = 1.9980 * ''f''(10000) = 1.9998 As ''x'' becomes extremely large, ''f''(''x'') approaches 2. In this case, :$\backslash lim\_\{x\; \backslash to\; \backslash infty\}\; f(x)\; =\; 2$ If one considers the codomain of ''f'' is the extension real line, then limit of a function at infinity could be considered as a special case of limit of a function at a point. :$\backslash lim\_\{x\; \backslash to\; \backslash infty\}\; f(x)\; =\; c$ if and only if for each $\backslash epsilon\; >\; 0\; \backslash quad\backslash exists\; n$ such that whenever $x\; >\; n$

Limit of a sequence
''Main article: limit of a sequence'' Consider the following sequence: 1.79, 1.799, 1.7999,... We could observe that the numbers are "approaching" 1.8, the limit of the sequence. Formally, suppose ''x''₁, ''x''₂, ... is a sequence of real numbers. We say that the real number ''L'' is the ''limit'' of this sequence and we write :$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; x\_n\; =\; L$ if and only if :for every ε>0 there exists a natural number ''n''₀ (which will depend on ε) such that for all ''n''>''n''₀ we have |''x''_''n'' - ''L''| < ε. Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit, since the absolute value |''x''_''n'' - ''L''| can be interpreted as the "distance" between ''x''_''n'' and ''L''. Not every sequence has a limit; if it does, we call it ''convergent'', otherwise ''divergent''. One can show that a convergent sequence has only one limit. The limit of a sequence and the limit of a function are closely related. On one hand, the limit of a sequence is simply the limit at infinity of a function defined on natural numbers. On the other hand, a limit of a function ''f'' at ''x'', if it exists, is the same as the limit of the sequence ''x''_''n''=''f''(''x''+1/''n'').

Topological net
''Main article: net (topology)'' All of the above notions of limit can be unified and generalized to arbitrary topological spaces by introducing topological net (topology) nets and defining their limits. The article on nets elaborates on this. An alternative is the concept of limit for Filter (mathematics) filters on topological spaces.

Limit in category theory
''Main article: limit (category theory)''

See also
*L'Hôpital's rule Category:Topology Category:Real analysis Category:Calculus {{Link FA|it}} ca:Límit cs:Limita de:Limes (Mathematik) es:Límite matemático eo:Limeso fr:Limite (mathématiques) ko:극한 io:Limito id:Limit it:Limite (matematica) he:גבול (מתמטיקה) lmo:Límit (matemàtica) hu:Határérték nl:Limiet ja:極é™? no:Grenseverdi pl:Granica (matematyka) pt:Limite fi:Raja-arvo sv:Gränsvärde tr:Limit uk:ГраницÑ? zh:æž?é™?

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