Definisi limit (ε, δ): Perbedaan antara revisi
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{{Short description|Definisi matematis limit}}{{Under construction}}{{DISPLAYTITLE:(''ε'', ''δ'')-definisi limit}}
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Dalam [[kalkulus]], '''definisi limit-(''ε'', ''δ'')''' (dibaca "definisi limit [[epsilon]]–[[delta (huruf)|delta]]) adalah formalisasi dari pengertian [[Limit fungsi|limit]]. Konsep tersebut karena [[Augustin-Louis Cauchy]], yang tidak pernah memberi nilai definisi limit (<math>\varepsilon,\delta</math>) dalam ''[[Cours d'Analyse]]'', tetapi terkadang digunakan argumen <math>\varepsilon,\delta</math> dalam bukti. Ini pertama kali diberikan sebagai definisi formal oleh [[Bernard Bolzano]] pada tahun 1817, dan pernyataan modern yang pasti akhirnya diberikan oleh [[Karl Weierstrass]].<ref name="grabiner">
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== Perbandingan dengan definisi infinitesimal ==
[[Howard Jerome Keisler|Keisler]] proved that a [[Hyperreal numbers|hyperreal]] [[Non-standard calculus#Limit|definition of limit]] reduces the [[logical quantifier]] complexity by two quantifiers.<ref>{{citation|last1=Keisler|first1=H. Jerome|chapter=Quantifiers in limits|title=Andrzej Mostowski and foundational studies|pages=151–170|publisher=IOS, Amsterdam|year=2008|contribution-url=http://www.math.wisc.edu/~keisler/limquant7.pdf}}</ref> Namely, <math>f(x)</math> converges to a limit ''L'' as <math>x</math> tends to ''a'' [[if and only if]] the value <math>f(x+e)</math> is infinitely close to ''L'' [[for every]] infinitesimal ''e''. (See [[Microcontinuity]] for a related definition of continuity, essentially due to [[Augustin-Louis Cauchy|Cauchy]].)
Infinitesimal calculus textbooks based on [[Abraham Robinson|Robinson]]'s approach provide definitions of continuity, derivative, and integral at standard points in terms of infinitesimals. Once notions such as continuity have been thoroughly explained via the approach using microcontinuity, the epsilon–delta approach is presented as well. [[Karel Hrbáček]] argues that the definitions of continuity, derivative, and integration in Robinson-style non-standard analysis must be grounded in the ''ε''–''δ'' method, in order to cover also non-standard values of the input.<ref>{{citation|last1=Hrbacek|first1=K.|editor-last=Van Den Berg|editor-first=I.|editor2-last=Neves|editor2-first=V.|chapter=Stratified Analysis?|title=The Strength of Nonstandard Analysis|publisher=Springer|year=2007}}</ref> Błaszczyk et al. argue that [[microcontinuity]] is useful in developing a transparent definition of uniform continuity, and characterize the criticism by Hrbáček as a "dubious lament".<ref>{{citation|last1=Błaszczyk|first1=Piotr|last2=Katz|first2=Mikhail|author2-link=Mikhail Katz|last3=Sherry|first3=David|arxiv=1202.4153|doi=10.1007/s10699-012-9285-8|journal=[[Foundations of Science]]|pages=43–74|title=Ten misconceptions from the history of analysis and their debunking|volume=18|year=2012|bibcode=2012arXiv1202.4153B|s2cid=119134151}}</ref> Hrbáček proposes an alternative non-standard analysis, which (unlike Robinson's) has many "levels" of infinitesimals, so that limits at one level can be defined in terms of infinitesimals at the next level.<ref>{{cite journal|last1=Hrbacek|first1=K.|year=2009|title=Relative set theory: Internal view|url=http://logicandanalysis.org/index.php/jla/article/view/25/17|journal=Journal of Logic and Analysis|volume=1}}</ref>
== Keluarga definisi limit formal ==
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