Editing Limit of a function (section)

==History==
Although implicit in the [[History of calculus|development of calculus]] of the 17th and 18th centuries, the modern idea of the limit of a function goes back to [[Bernard Bolzano]] who, in 1817, introduced the basics of the epsilon-delta technique (see [[#(ε, δ)-definition of limit|(ε, δ)-definition of limit]] below) to define continuous functions. However, his work was not known during his lifetime.<ref>{{Citation|title=Bolzano, Cauchy, Epsilon, Delta|last=Felscher|first=Walter|journal=American Mathematical Monthly|volume=107|issue=9|pages=844–862|year=2000|doi=10.2307/2695743|jstor=2695743}}</ref> Bruce Pourciau argues that [[Isaac Newton]], in his 1687 ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'', demonstrates a more sophisticated understanding of limits than he is generally given credit for, including being the first to present an epsilon argument.<ref>{{Cite journal |last=Pourciau |first=Bruce |date=2001 |title=Newton and the Notion of Limit |url=https://linkinghub.elsevier.com/retrieve/pii/S0315086000923012 |journal=Historia Mathematica |language=en |volume=28 |issue=1 |pages=18–30 |doi=10.1006/hmat.2000.2301}}</ref><ref>{{Cite journal |last=Pourciau |first=Bruce |date=2009 |title=Proposition II (Book I) of Newton's "Principia" |url=https://www.jstor.org/stable/41134303 |journal=Archive for History of Exact Sciences |volume=63 |issue=2 |pages=129–167 |doi=10.1007/s00407-008-0033-y |jstor=41134303 |issn=0003-9519}}</ref>

In his 1821 book {{lang|fr|[[Cours d'analyse]]}}, [[Augustin-Louis Cauchy]] discussed variable quantities, [[infinitesimal]]s and limits, and defined continuity of <math>y=f(x)</math> by saying that an infinitesimal change in {{mvar|x}} necessarily produces an infinitesimal change in {{mvar|y}}, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs.<ref name="Grabiner1983">{{Citation
| last = Grabiner | first = Judith V.
| title = Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
| journal = [[American Mathematical Monthly]]
| volume = 90
| issue = 3
| pages = 185–194
| doi = 10.2307/2975545
| year = 1983
| jstor = 2975545
}}, collected in [http://www.maa.org/ebooks/spectrum/WGE.html Who Gave You the Epsilon?], {{isbn|978-0-88385-569-0}} pp. 5–13. Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf</ref> In 1861, [[Karl Weierstrass]] first introduced the epsilon-delta definition of limit in the form it is usually written today.<ref>{{citation
 | last1 = Sinkevich |first1 = G. I.
 | title = Historia epsylontyki
 | journal = [[Antiquitates Mathematicae]]
 | year = 2017
 | volume = 10
 | publisher = Cornell University
 | doi = 10.14708/am.v10i0.805
 | arxiv = 1502.06942}}</ref> He also introduced the notations <math display="inline">\lim</math> and <math display="inline">\textstyle \lim_{x \to x_0} \displaystyle.</math><ref>{{citation
 | last = Burton | first = David M.
 | title = The History of Mathematics: An introduction
 | edition = Third
 | publisher = McGraw–Hill
 | location = New York
 | year = 1997
 | pages = 558–559
 | isbn = 978-0-07-009465-9}}</ref>

The modern notation of placing the arrow below the limit symbol is due to [[G. H. Hardy]], which is introduced in his book ''[[A Course of Pure Mathematics]]'' in 1908.<ref>{{citation
 | last = Miller | first = Jeff
 | title = Earliest Uses of Symbols of Calculus
 | date = 1 December 2004
 | url = http://jeff560.tripod.com/calculus.html
 | access-date = 2008-12-18}}</ref>