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| {{dablink|This is an overview of the idea of a limit in mathematics. For specific uses of a limit, see [[Limit of a sequence]] and [[Limit of a function]].}}
| | There are many online companies that write custom scholarship essays for students. You will want to map out exactly how you want your work to flow, to ensure that it makes sense. They will be good with the researching and coming up with creative ideas but not the actual writing process. The organization and process of writing requires planning, comprehensive research, research, secondary research and research before beginning writing. This is where to expound on your arguments and defend your thesis statement by stating facts or theories, plus your own insights on why you find them so. <br><br>Definition essay- A definition essay defines a particular term. This will make your life much easier, and your teacher or professor will love you for not having to strain to read what you attempted to squeeze between lines in illegible scribble. Custom essay writing services today can produce anything ranging from originally researched and written term papers, theses and essays to articles and blogs for people, organizations, websites and individuals based on their needs and requirements. This will also give coherence to a comparison paragraph. Our experts also impart variety of services like Custom Essay Writing, Critical Essay Writing, Essay Project Help, Essay writing Help. <br><br>Since you were admitted for college, you may have been missing free time to engage with other activities simply because of the hardships in your writings. Descriptive essays, as the name indicate are used to provide a vivid description of a person, place or thing. This makes writing argumentative essay a very difficult task. A reputable company hires qualified writers with different fields of specialization to cater for writing a variety of essays depending upon the subject to which the topic relates. When such a company must write my essay for me I make certain that I buy the best offer. <br><br>Our custom analytical essay will provide you with enough time to attend to other duties. The followings describe what topics you are ready to obtain when you're looking to buy essay paper. The introduction section of an essay consists of the main idea of the essay. And they can ask the kinds of quirky questions'both short answer and full-on essays'that require an element of self-reflection on the part of the applicant. Selection of catchy and precise title will definitely improve the quality of essay. <br><br>If you have any inquiries concerning in which and how to use [http://dracula.godohosting.com/xe/?document_srl=31350 do my essay cheap], you can get hold of us at the site. In this paragraph you would simply have to retell the hypothesis which you have mentioned earlier. Are you looking for excellent value in movies online. When defining your thesis, enclose it in one statement. Soon after writing the plan, you need to head to the library to find the books from the reading list or search for journal articles on the internet. Certainly while drafting essays, the first thought that will come to your mind is the latest event that you have experienced. |
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| In [[mathematics]], a '''limit''' is the value that a [[function (mathematics)|function]] or [[sequence]] "approaches" as the input or index approaches some value.<ref>{{cite book|last=Stewart|first=James|authorlink=James Stewart (mathematician)|title=Calculus: Early Transcendentals|publisher=[[Brooks/Cole]]|edition=6th|year=2008|isbn =0-495-01166-5}}</ref> Limits are essential to [[calculus]] (and [[mathematical analysis]] in general) and are used to define [[continuous function|continuity]], [[derivative]]s, and [[integral]]s.
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| The concept of a [[limit of a sequence]] is further generalized to the concept of a limit of a [[net (topology)|topological net]], and is closely related to [[limit (category theory)|limit]] and [[direct limit]] in [[category theory]].
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| In formulas, a limit is usually denoted "lim" as in {{nowrap|lim<sub>''n'' → ''c''</sub>(''a''<sub>''n''</sub>) {{=}} ''L''}}, and the fact of approaching a limit is represented by the right arrow (→) as in ''a''<sub>''n''</sub> → ''L''.
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| == Limit of a function ==
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| {{main|Limit of a function}}
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| {{Double image|right|Límite 01.svg|{{#expr: (200 * (800 / 800)) round 0}}|Limit-at-infinity-graph.png|{{#expr: (200 * (619 / 405)) round 0}}|Whenever a point {{math|x}} is within δ units of {{math|c}}, {{math|f(x)}} is within ε units of {{math|L}}.|For all {{math|x > S}}, {{math|f(x)}} is within ε of {{math|L}}.}}
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| Suppose {{math|f}} is a [[real-valued function]] and {{math|c}} is a [[real number]]. The expression
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| :<math> \lim_{x \to c}f(x) = L </math>
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| means that {{math|f(x)}} can be made to be as close to {{math|L}} as desired by making {{math|x}} sufficiently close to {{math|c}}. In that case, the above equation can be read as "the limit of {{math|f}} of {{math|x}}, as {{math|x}} approaches {{math|c}}, is {{math|L}}".
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| [[Augustin-Louis Cauchy]] in 1821,<ref name=Larson>{{Cite book|first1=Ron|last1=Larson|authorlink1=Ron Larson (mathematician)|first2=Bruce H.|last2=Edwards|title=Calculus of a single variable|edition=Ninth|publisher=[[Brooks/Cole]], [[Cengage Learning]]|year=2010|isbn=978-0-547-20998-2}}</ref> followed by [[Karl Weierstrass]], formalized the definition of the limit of a function as the above definition, which became known as the [[(ε, δ)-definition of limit]] in the 19th century. The definition uses {{math|[[ε]]}} (the lowercase Greek letter ''epsilon'') to represent a small positive number, so that "{{math|f(x)}} becomes arbitrarily close to {{math|L}}" means that {{math|f(x)}} eventually lies in the interval {{math|(L - ε, L + ε)}}, which can also be written using the absolute value sign as {{math|{{!}}f(x) - L{{!}} < ε}}.<ref name=Larson/> The phrase "as {{math|x}} approaches {{math|c}}" then indicates that we refer to values of {{math|x}} whose distance from {{math|c}} is less than some positive number {{math|[[δ]]}} (the lower case Greek letter ''delta'')—that is, values of {{math|x}} within either {{math|(c - δ, c)}} or {{math|(c, c + δ)}}, which can be expressed with {{math|0 < {{!}}x - c{{!}} < δ}}. The first inequality means that the distance between {{math|x}} and {{math|c}} is greater than 0 and that {{math|x ≠ c}}, while the second indicates that {{math|x}} is within distance {{math|δ}} of {{math|c}}.<ref name=Larson/>
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| Note that the above definition of a limit is true even if {{math|f(c) ≠ L}}. Indeed, the function {{math|f}} need not even be defined at {{math|c}}.
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| For example, if
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| :<math> f(x) = \frac{x^2 - 1}{x - 1} </math>
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| then ''f''(1) is not defined (see [[division by zero]]), yet as {{math|x}} moves arbitrarily close to 1, {{math|f(x)}} correspondingly approaches 2:
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| {| class="wikitable"
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| |''f''(0.9)||''f''(0.99)||''f''(0.999)|| ''f''(1.0) ||''f''(1.001)||''f''(1.01)||''f''(1.1)
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| |-
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| | 1.900 || 1.990 || 1.999 || ⇒ undefined ⇐ || 2.001 || 2.010 || 2.100
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| |}
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| Thus, {{math|f(x)}} can be made arbitrarily close to the limit of 2 just by making {{math|x}} sufficiently close to 1.
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| In other words, <math> \lim_{x \to 1} \frac{x^2-1}{x-1} = 2 </math>
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| This can also be calculated algebraically, as <math>\frac{x^2-1}{x-1} = \frac{(x+1)(x-1)}{x-1} = x+1</math> for all real numbers <math>x\neq 1</math>.
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| Now since <math>x+1</math> is continuous in <math>x</math> at 1, we can now plug in 1 for <math>x</math>, thus <math>\lim_{x \to 1} \frac{x^2-1}{x-1} = 1+1 = 2</math>.
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| In addition to limits at finite values, functions can also have limits at infinity. For example, consider
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| :<math>f(x) = {2x-1 \over x}</math>
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| * ''f''(100) = 1.9900
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| * ''f''(1000) = 1.9990
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| * ''f''(10000) = 1.99990
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| As {{math|x}} becomes extremely large, the value of {{math|f(x)}} approaches 2, and the value of {{math|f(x)}} can be made as close to 2 as one could wish just by picking {{math|x}} sufficiently large. In this case, the limit of {{math|f(x)}} as {{math|x}} approaches infinity is 2. In mathematical notation,
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| :<math> \lim_{x \to \infty} \frac{2x-1}{x} = 2. </math>
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| == Limit of a sequence ==
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| {{main|Limit of a sequence}}
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| Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are "approaching" 1.8, the limit of the sequence.
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| Formally, suppose ''a''<sub>1</sub>, ''a<sub>2</sub>'', ... is a [[sequence]] of [[real number]]s. It can be stated that the real number {{math|L}} is the ''limit'' of this sequence, namely:
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| :<math> \lim_{n \to \infty} a_n = L </math>
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| to mean
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| :For every [[real number]] ε > 0, there exists a [[natural number]] ''n''<sub>0</sub> such that for all ''n'' > ''n''<sub>0</sub>, |''a''<sub>''n''</sub> − {{math|L}}| < ε.
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| Intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the [[absolute value]] |''a''<sub>''n''</sub> − {{math|L}}| is the distance between ''a''<sub>''n''</sub> and {{math|L}}. Not every sequence has a limit; if it does, it is called ''[[Convergent series|convergent]]'', and if it does not, it is ''divergent''. One can show that a convergent sequence has only one limit.
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| The limit of a sequence and the limit of a function are closely related. On one hand, the limit as ''n'' goes to infinity of a sequence ''a''(''n'') is simply the limit at infinity of a function defined on the [[natural number]]s ''n''. On the other hand, a limit {{math|L}} of a function ''f''({{math|x}}) as {{math|x}} goes to infinity, if it exists, is the same as the limit of any arbitrary sequence ''a<sub>n</sub>'' that approaches {{math|L}}, and where ''a<sub>n</sub>'' is never equal to {{math|L}}. Note that one such sequence would be {{nowrap|{{math|L}} + 1/''n''}}. | |
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| ==Limit as "standard part"==
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| In [[non-standard analysis]] (which involves a [[hyperreal number|hyperreal]] enlargement of the number system), the limit of a sequence <math>(a_n)</math> can be expressed as the [[standard part function|standard part]] of the value <math>a_H</math> of the natural extension of the sequence at an infinite [[hypernatural]] index ''n=H''. Thus,
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| :<math> \lim_{n \to \infty} a_n = \operatorname{st}(a_H) </math>.
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| Here the standard part function "st" rounds off each finite hyperreal number to the nearest real number (the difference between them is [[infinitesimal]]). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal <math>a=[a_n]</math> represented in the ultrapower construction by a Cauchy sequence <math>(a_n)</math>, is simply the limit of that sequence:
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| :<math> \operatorname{st}(a)=\lim_{n \to \infty} a_n </math>.
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| In this sense, taking the limit and taking the standard part are equivalent procedures.
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| ==Convergence and fixed point==
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| A formal definition of convergence can be stated as follows.
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| Suppose <math> { {p}_{n} } </math> as <math> n </math> goes from <math> 0 </math> to <math> \infty </math> is a sequence that converges to <math> p </math>, with <math> {p}_{n} \neq p </math> for all <math> n </math>. If positive constants <math> \lambda </math> and <math> \alpha </math> exist with
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| ::::::<math>\lim_{n \rightarrow \infty } \frac{ \left | { p}_{n+1 } -p \right | }{ { \left | { p}_{n }-p \right | }^{ \alpha} } =\lambda </math>
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| then <math> { {p}_{n} } </math> as <math> n </math> goes from <math> 0 </math> to <math> \infty </math> converges to <math> p </math> of order <math> \alpha </math>, with asymptotic error constant <math> \lambda </math>
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| Given a function <math> f </math> with a fixed point <math> p </math>, there is a nice checklist for checking the convergence of the sequence <math>p_n</math>.
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| :1) First check that p is indeed a fixed point: | |
| ::<math> f(p) = p </math>
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| :2) Check for linear convergence. Start by finding <math>\left | f^\prime (p) \right | </math>. If....
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| {| class="wikitable" border="1"
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| | <math>\left | f^\prime (p) \right | \in (0,1)</math>
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| | then there is linear convergence
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| |-
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| | <math>\left | f^\prime (p) \right | > 1</math>
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| | series diverges
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| |-
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| | <math>\left | f^\prime (p) \right | =0 </math>
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| | then there is at least linear convergence and maybe something better, the expression should be checked for quadratic convergence
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| |}
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| :3) If it is found that there is something better than linear the expression should be checked for quadratic convergence. Start by finding <math>\left | f^{\prime\prime} (p) \right | </math> If....
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| {| class="wikitable" border="1"
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| | <math>\left | f^{\prime\prime} (p) \right | \neq 0</math>
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| | then there is quadratic convergence provided that <math> f^{\prime\prime} (p) </math> is continuous
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| |-
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| | <math>\left | f^{\prime\prime} (p) \right | = 0</math>
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| | then there is something even better than quadratic convergence
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| |-
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| | <math>\left | f^{\prime\prime} (p) \right | </math> does not exist
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| | then there is convergence that is better than linear but still not quadratic
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| |}
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| <ref>''Numerical Analysis'', 8th Edition, Burden and Faires, Section 2.4 Error Analysis for Iterative Methods </ref>
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| == Topological net ==
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| {{main|Net (topology)}}
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| <!-- Better introduction is needed -->
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| All of the above notions of limit can be unified and generalized to arbitrary [[topological space]]s by introducing topological [[net (topology)|nets]] and defining their limits.
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| An alternative is the concept of limit for [[Filter (mathematics)|filters]] on topological spaces.
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| == See also ==
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| {{wikibooks|Calculus|Limits}}
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| *[[Limit of a sequence]]
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| **[[Rate of convergence]]: the rate at which a convergent sequence approaches its limit
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| *[[Cauchy sequence]]
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| **[[complete metric space]]
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| *[[Limit of a function]]
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| **[[One-sided limit]]: either of the two limits of functions of a real variable ''x'', as ''x'' approaches a point from above or below
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| **[[List of limits]]: list of limits for common functions
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| **[[Squeeze theorem]]: finds a limit of a function via comparison with two other functions
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| *[[Banach limit]] defined on the Banach space that extends the usual limits.
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| *[[Limit (category theory)|Limit in category theory]]
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| **[[Direct limit]]
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| **[[Inverse limit]]
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| *[[Asymptotic analysis]]: a method of describing limiting behavior
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| **[[Big O notation]]: used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity
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| *[[Convergent matrix]]
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| ==Notes==
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| {{reflist}}
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| == External links ==
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| {{Library resources box
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| |by=no
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| |onlinebooks=no
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| |others=no
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| |about=yes
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| |label=Limit (mathematics)}}
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| * {{MathWorld |title=Limit |urlname=Limit}}
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| * [http://www.mathwords.com/l/limit.htm Mathwords: Limit]
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| <!-- Limits here look all right, i can't guarantee for all site's content. -->
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| [[Category:Limits (mathematics)| ]]
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| [[Category:Real analysis]]
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| [[Category:Asymptotic analysis]]
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| [[Category:Differential calculus]]
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| [[Category:General topology]]
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| {{Link FA|lmo}}
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