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{{Periksa terjemahan}}
 
{{Short description|Definisi matematis limit}}
{{DISPLAYTITLE:Definisi limit (''{{mvar|ε'', ''δ''}})-definisi limit}}
{{periksaterjemahan|en|(ε, δ)-definition of limit}}
[[Berkas:Límite 01.svg|thumb|right|Kapanpun suatu titik ''x'' is within ''δ'' unit ''c'', ''f''(''x'') berada dalam ε unit ''L'']]
[[Berkas:Límite 01.svg|thumb|right|Apabila titik <math>x</math> berada di satuan <math>\delta</math> dari <math>c</math>, <math>f(x)</math> berada di satuan <math>\varepsilon</math> dari <math>L</math>]]
 
Dalam [[kalkulus]], '''definisi limit-(''ε'',&nbsp; ''δ'')-definisi limit''' (dibaca "definisi limit [[epsilon]]–[[delta (huruf)|delta]] definisi limit") adalah formalisasi dari pengertian [[Limit fungsi|limit]]. Konsep tersebut karena [[Augustin-Louis Cauchy]], yang tidak pernah memberi nilai definisi limit (<math>\varepsilon,\delta</math>) definisi batas dalam ''[[Cours d'Analyse]]'', tetapi kadang-kadangterkadang digunakan argumen <math>\varepsilon,\delta</math> argumen dalam bukti. Ini pertama kali diberikan sebagai definisi formal oleh [[Bernard Bolzano]] pada tahun 1817, dan pernyataan modern yang definitifpasti akhirnya diberikan oleh [[Karl Weierstrass]].<ref name="grabiner">
{{citation
|title=Siapa yang Memberi Anda Epsilon? Cauchy dan Origins of Rigorous Calculus
Baris 39 ⟶ 38:
|access-date = 2009-05-01
|df =
}}. Accessed 2009-05-01.</ref> Hal tersebut memberikan ketelitian pada gagasan informal berikut: ekspresiungkapan dependentergantung ''<math>f''(''x'')</math> mendekati nilai ''<math>L''</math>, sebagai variabel ''<math>x''</math> mendekati nilai ''<math>c'',</math> bilajika ''<math>f''(''x'')</math> dapat dibuat sedekat yang diinginkan ''<math>L''</math>, dengan mengambil nilai ''<math>x''</math> yang cukup dekat dengan nilai ''<math>c''</math>.
 
==Sejarah==
 
Meskipun orang Yunani memeriksa proses pembatasan, seperti [[metode Babilonia]], mereka mungkin tidak memiliki konsep yang mirip dengan modern limit.<ref>{{cite book|last1=Stillwell|first1=John|authorlink=John Stillwell|title=Matematika dan Sejarahnya|url=https://archive.org/details/mathematicsitshi0000stil|url-access=registration|date=1989|publisher=Springer-Verlag|location=New York|isbn=978-1-4899-0007-4|pages=[https://archive.org/details/mathematicsitshi0000stil/page/38 38–39]}}</ref> Ketentuan konsep limit muncul pada tahun 1600-an, ketika [[Pierre de Fermat]] berusaha menemukan [[kemiringankelerengan]] dari garis [[tangen]] pada suatu titik <math>x</math> dari fungsi seperti <math>f(x)=x^2</math>. Menggunakan kuantitas bukan noltaknol tetapi hampir nol, <math>E</math>, Fermat melakukan perhitungan berikut:
 
:<math>
\begin{align}
\text{lerengankelerengan} & = \frac{f(x+E)-f(x)}{E} \\
& = \frac{(x+E)^2-x^2}{E}\\
& = \frac{x^2+2xE+E^2-x^2}{E} \\
& = \frac{2xE+E^2}{E} = 2x+E = 2x.
\end{align}
</math>
 
Kunci dari perhitungan di atas adalah sejak <math>E</math> bukan noltaknol, seseorangsalah satunya dapat membagi <math>f(x+E)-f(x)</math> dari <math>E</math>, tapi sejakketika <math>E</math> dekat dengan <math>0</math>, <math>2x+E</math> pada dasarnya adalah <math>2x</math>.<ref>{{cite book|last1=Stillwell|first1=John|authorlink=John Stillwell|title=Matematika dan Sejarahnya|url=https://archive.org/details/mathematicsitshi0000stil|url-access=registration|date=1989|publisher=Springer-Verlag|location=New York|isbn=978-1-4899-0007-4|pages=[https://archive.org/details/mathematicsitshi0000stil/page/104 104]}}</ref> Kuantitas seperti <math>E</math> disebut [[infinitesimal]]. Masalah dengan kalkulasiperhitungan ini adalah bahwa para matematikawan zaman itu tidak dapat secara tepat mendefinisikan kuantitas dengan sifat <math> E</math>,<ref>{{cite book|last1=Stillwell|first1=John|authorlink=John Stillwell|title=Matematika dan Sejarahnya|url=https://archive.org/details/mathematicsitshi0000stil|url-access=registration|date=1989|publisher=Springer-Verlag|location=New York|isbn=978-1-4899-0007-4|pages=[https://archive.org/details/mathematicsitshi0000stil/page/106 106]}}</ref>, meskipun itu adalah praktik umum untuk 'mengabaikan' kekuatan takinfinitesimal terbataspangkat yang lebih tinggi dan ini tampaknya membuahkan hasil yang benar.
 
Masalah ini muncul kembali kemudian pada tahun 1600-an1600an di pusat perkembangan [[kalkulus]], karena perhitungan seperti Fermat penting untuk perhitungan [[turunan]]. [[Isaac Newton]] kalkulus yang dikembangkan pertama kali melalui jumlah yang sangat kecil yang disebut [[Metode Fluks|fluks]]. Dia mengembangkannya dengan mengacu pada gagasan tentang "momen waktu yang sangat kecil..."<ref name="ReferenceA">{{cite book|last1=Buckley|first1=Benjamin Lee|title=Perdebatan kontinuitas: Dedekind, Cantor, du Bois-Reymond dan Peirce tentang kontinuitas dan infinitesimal|date=2012|isbn=9780983700487|page=31}}</ref> Namun, Newton kemudian menolak fluks demi teori rasio yang mendekati modern <math>\varepsilon\text{–}\delta </math> definisi bataslimit.<ref name="ReferenceA"/> Selain itu, Newton menyadari bahwa bataslimit rasio jumlah yangkuantitas hilanglenyap adalah ''bukan'' rasio itu sendiri<!--, saat ia menulis:
:Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...
Additionally, Newton occasionally explained limits in terms similar to the epsilon–delta definition.<ref>{{citation|title=Newton and the Notion of Limit| first1=B.|last1=Pourciau|journal=Historia Mathematica|volume=28|issue=1| pages=18–30|year=2001|doi=10.1006/hmat.2000.2301 }}</ref> [[Gottfried Wilhelm Leibniz]] developed an infinitesimal of his own and tried to provide it with a rigorous footing, but it was still greeted with unease by some mathematicians and philosophers.<ref>{{cite book|last1=Buckley|first1=Benjamin Lee|title=The continuity debate : Dedekind, Cantor, du Bois-Reymond and Peirce on continuity and infinitesimals|date=2012|isbn=9780983700487|page=32}}</ref>
 
: Rasio terakhirnya ... sebenarnya bukan rasio kauntitas terakhirnya, tetapi limit ... yang mana ini dapat didekatkan lebih dekat bahwa perbedaannya lebih kecil dari suatu kuantitas yang diberikan...
[[Augustin-Louis Cauchy]] gave a definition of limit in terms of a more primitive notion he called a ''variable quantity''. He never gave an epsilon–delta definition of limit (Grabiner 1981). Some of Cauchy's proofs contain indications of the epsilon–delta method. Whether or not his foundational approach can be considered a harbinger of Weierstrass's is a subject of scholarly dispute. Grabiner feels that it is, while Schubring (2005) disagrees.{{dubious|date=December 2011}}<ref name="grabiner" /> Nakane concludes that Cauchy and Weierstrass gave the same name to different notions of limit.<ref>Nakane, Michiyo. Did Weierstrass's differential calculus have a limit-avoiding character? His definition of a limit in ''ε''−''δ'' style. BSHM Bull. 29 (2014), no. 1, 51–59.</ref>{{Unreliable source?|date=April 2015}}
 
Sebagai tambahan, Newton terkadang menjelaskan limit dalam istilah yang serupa dengan definisi epsilon–delta.<ref>{{citation|title=Newton and the Notion of Limit|first1=B.|last1=Pourciau|journal=Historia Mathematica|volume=28|issue=1|pages=18–30|year=2001|doi=10.1006/hmat.2000.2301}}</ref> [[Gottfried Wilhelm Leibniz]] mengembangkan sebuah infinitesimal oleh dirinya dan mencoba untuk memberikannya dengan sebuah dasar yang setepat-tepatnya, tetapi ini tetap disambut dengan gelisah oleh beberapa matematikawan dan para filsafat.<ref>{{cite book|last1=Buckley|first1=Benjamin Lee|date=2012|title=The continuity debate : Dedekind, Cantor, du Bois-Reymond and Peirce on continuity and infinitesimals|isbn=9780983700487|page=32}}</ref>
Eventually, Weierstrass and Bolzano are credited with providing a rigorous footing for calculus, in the form of the modern <math>\varepsilon\text{–}\delta</math> definition of the limit.<ref name="grabiner"/><ref>{{citation
|first = A.-L.
|last = Cauchy
|author-link = Augustin Louis Cauchy
|title = Résumé des leçons données à l'école royale polytechnique sur le calcul infinitésimal
|place = Paris
|year = 1823
|url = http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0
|chapter = Septième Leçon - Valeurs de quelques expressions qui se présentent sous les formes indéterminées <math>\frac{\infty}{\infty}, \infty^0, \ldots</math> Relation qui existe entre le rapport aux différences finies et la fonction dérivée
|chapter-url = http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f45n5.capture
|postscript = , [http://gallica.bnf.fr/ark:/12148/bpt6k90196z.image.f47 p. 44].
|archive-url = https://www.webcitation.org/5gVUmywgY?url=http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0
|archive-date = 2009-05-04
|url-status = dead
|access-date = 2009-05-01
|df =
}}.</ref> The need for reference to an infinitesimal <math>E</math> was then removed,<ref>{{cite book|last1=Buckley|first1=Benjamin Lee|title=The continuity debate : Dedekind, Cantor, du Bois-Reymond and Peirce on continuity and infinitesimals|date=2012|isbn=9780983700487|page=33}}</ref> and Fermat's computation turned into the computation of the following limit:
 
[[Augustin-Louis Cauchy]] memberikan sebuah definisi limit dalam hal gagasan lebih primitif yang disebut sebuah ''kuantitas variabel''. Dia tidak pernah memberikan epsilon–delta definisi limit (Grabiner 1981). Beberapa bukti Cauchy berisi indikasi metode epsilon–delta. Whether or not his foundational approach can be considered a harbinger of Weierstrass's is a subject of scholarly dispute. Grabiner feels that it is, while Schubring (2005) disagrees.{{dubious|date=December 2011}}<ref name="grabiner" /> Nakane concludes that Cauchy and Weierstrass gave the same name to different notions of limit.<ref>Nakane, Michiyo. Did Weierstrass's differential calculus have a limit-avoiding character? His definition of a limit in ''ε''−''δ'' style. BSHM Bull. 29 (2014), no. 1, 51–59.</ref>{{Unreliable source?|date=April 2015}}
:<math>
 
Eventually, Weierstrass and Bolzano are credited with providing a rigorous footing for calculus, in the form of the modern <math>\varepsilon\text{–}\delta</math> definition of the limit.<ref name="grabiner" /><ref>{{citation|first=A.-L.|last=Cauchy|author-link=Augustin Louis Cauchy|title=Résumé des leçons données à l'école royale polytechnique sur le calcul infinitésimal|place=Paris|year=1823|url=http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0|chapter=Septième Leçon - Valeurs de quelques expressions qui se présentent sous les formes indéterminées <math>\frac{\infty}{\infty}, \infty^0, \ldots</math> Relation qui existe entre le rapport aux différences finies et la fonction dérivée|chapter-url=http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f45n5.capture|postscript=, [http://gallica.bnf.fr/ark:/12148/bpt6k90196z.image.f47 p. 44].|archive-url=https://www.webcitation.org/5gVUmywgY?url=http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0|archive-date=2009-05-04|url-status=dead|access-date=2009-05-01|df=}}.</ref> The need for reference to an infinitesimal <math>E</math> was then removed,<ref>{{cite book|last1=Buckley|first1=Benjamin Lee|date=2012|title=The continuity debate : Dedekind, Cantor, du Bois-Reymond and Peirce on continuity and infinitesimals|isbn=9780983700487|page=33}}</ref> and Fermat's computation turned into the computation of the following limit:
 
: <math>
\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}.
</math>
 
This is not to say that the limiting definition was free of problems as, although it removed the need for infinitesimals, it did require the construction of the [[realReal number|real numbers]]s by [[Richard Dedekind]].<ref>{{cite book|last1=Buckley|first1=Benjamin Lee|date=2012|title=The continuity debate : Dedekind, Cantor, du Bois-Reymond and Peirce on continuity and infinitesimals|date=2012|isbn=9780983700487|pages=32–35}}</ref> This is also not to say that infinitesimals have no place in modern mathematics, as later mathematicians were able to rigorously create infinitesimal quantities as part of the [[hyperreal number]] or [[surreal number]] systems. Moreover, it is possible to rigorously develop calculus with these quantities and they have other mathematical uses.<ref>{{cite book|last1=Tao|first1=Terence|date=2008|url=https://archive.org/details/structurerandomn00taot|title=Structure and randomness : pages from year one of a mathematical blog|date=2008|publisher=American Mathematical Society|location=Providence, R.I.|publisher=American Mathematical Society|isbn=978-0-8218-4695-7|pages=95–110[https://archive.org/details/structurerandomn00taot/page/95 95]–110}}</ref>-->
 
==Pernyataan informal==
- Dalam pengembangan -
<!-- [Bagian ini sedang dalam terjemahan] A viable informal (that is, intuitive or provisional) definition is that a "[[Function (mathematics)|function]] ''f'' approaches the limit ''L'' near ''a'' (symbolically, <math> \lim_{x \to a}f(x) = L \, </math>) if we can make ''f''(''x'') as close as we like to ''L'' by requiring that ''x'' be sufficiently close to, but unequal to, ''a''."<ref>{{cite book|last1=Spivak|first1=Michael|title=Calculus|url=https://archive.org/details/calculus4thediti00mich|url-access=registration|date=2008|publisher=Publish or Perish|location=Houston, Tex.|isbn=978-0914098911|page=[https://archive.org/details/calculus4thediti00mich/page/90 90]|edition=4th}}</ref>
 
When we say that two things are close (such as ''f''(''x'') and ''L'' or ''x'' and ''a''), we mean that the difference (or [[distance]]) between them is small. When ''f''(''x''), ''L'', ''x'', and ''a'' are [[real number]]s, the difference/distance between two numbers is the [[absolute value]] of the [[Subtraction|difference]] of the two. Thus, when we say ''f''(''x'') is close to ''L'', we mean that <math>|f(x)-L|</math> is small. When we say that ''x'' and ''a'' are close, we mean that <math> |x-a|</math> is small.<ref name="Calculus">{{cite book|last1=Spivak|first1=Michael|title=Calculus|url=https://archive.org/details/calculus4thediti00mich|url-access=registration|date=2008|publisher=Publish or Perish|location=Houston, Tex.|isbn=978-0914098911|page=[https://archive.org/details/calculus4thediti00mich/page/96 96]|edition=4th}}</ref>
 
When we say that we can make ''f''(''x'') as close as we like to ''L'', we mean that for '''all''' non-zero distances, <math>\varepsilon</math>, we can make the distance between ''f''(''x'') and ''L'' smaller than <math>\varepsilon</math>.<ref name="Calculus"/>
 
When we say that we can make ''f''(''x'') as close as we like to ''L'' by requiring that ''x'' be sufficiently close to, but, unequal to, ''a'', we mean that for every non-zero distance <math>\varepsilon </math>, there is some non-zero distance <math>\delta </math> such that if the distance between ''x'' and ''a'' is less than <math>\delta </math> then the distance between ''f(x)'' and ''L'' is smaller than <math>\varepsilon </math>.<ref name="Calculus"/>
 
The informal/intuitive aspect to be grasped here is that the definition requires the following internal conversation (which is typically paraphrased by such language as "your enemy/adversary attacks you with an <math> \epsilon </math>, and you defend/protect yourself with a <math> \delta </math>"): One is provided with any challenge <math> \epsilon > 0 </math> for a given ''f'',''a'', and ''L''. One must answer with a <math> \delta > 0 </math> such that <math> 0 < |x-a | < \delta </math> implies that <math> |f(x)-L| < \epsilon </math>. Both <math> \epsilon </math> and <math> \delta </math> are generally understood to be small quantities,<ref>{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/|access-date=2020-08-18|website=Math Vault|language=en-US}}</ref> and if one can provide an answer for any challenge, then one has proven that the limit exists.<ref>{{Cite web|title=Epsilon-Delta Definition of a Limit {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/epsilon-delta-definition-of-a-limit/|access-date=2020-08-18|website=brilliant.org|language=en-us}}</ref>-->
 
==Pernyataan yang tepat dan pernyataan terkait==
- Dalam pengembangan -
===Pernyataan yang tepat untuk fungsi bernilai nyata===
<!-- [Bagian ini sedang dalam terjemahan] The <math>(\varepsilon, \delta)</math> definition of the [[limit of a function]] is as follows:<ref name="Calculus"/>
 
Let <math>f</math> be a [[real-valued function]] defined on a subset <math>D</math> of the [[real number]]s. Let <math>c</math> be a [[limit point]] of <math>D</math> and let <math>L</math> be a real number. We say that
 
: <math> \lim_{x\to c}f(x) = L </math>
 
if for every <math> \varepsilon > 0 </math> there exists a <math> \delta > 0 </math> such that, for all <math>x\in D</math>, if <math> 0 < |x-c| < \delta </math>, then <math> |f(x)-L| < \varepsilon</math>.<ref>{{Cite web|date=2017-04-21|title=1.2: Epsilon-Delta Definition of a Limit|url=https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(Apex)/01%3A_Limits/1.02%3A_Epsilon-Delta_Definition_of_a_Limit|access-date=2020-08-18|website=Mathematics LibreTexts|language=en}}</ref>
 
Symbolically:
:<math> \lim_{x \to c} f(x) = L \iff (\forall \varepsilon > 0,\,\exists \ \delta > 0,\,\forall x \in D,\,0 < |x - c| < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon)</math>
 
If <math>D=[a,b]</math> or <math>D=\mathbb{R}</math>, then the condition that <math> c</math> is a limit point can be replaced with the simpler condition that ''c'' belongs to ''D'', since closed [[Interval (mathematics)|real intervals]] and the entire real line are [[perfect set]]s.-->
 
===Pernyataan yang tepat untuk fungsi antara ruang metrik===
 
<!--The definition can be generalized to functions that map between [[metric space]]s. These spaces come with a function, called a metric, that takes two points in the space and returns a real number that represents the distance between the two points.<ref name="Rudin, Walter 1976 30">{{cite book|author=Rudin, Walter|title=Principles of Mathematical Analysis|publisher= McGraw-Hill Science/Engineering/Math |year=1976|isbn= 978-0070542358 |page=[https://archive.org/details/principlesofmath00rudi/page/30 30]|url=https://archive.org/details/principlesofmath00rudi|url-access=registration}}</ref> The generalized definition is as follows:<ref>{{cite book|author=Rudin, Walter|title=Principles of Mathematical Analysis|publisher= McGraw-Hill Science/Engineering/Math |year=1976|isbn= 978-0070542358 |page=[https://archive.org/details/principlesofmath00rudi/page/83 83]|url=https://archive.org/details/principlesofmath00rudi|url-access=registration}}</ref>
 
Suppose <math>f</math> is defined on a subset <math>D</math> of a metric space <math>X</math> with a metric <math> d_X(x,y)</math> and maps into a metric space <math>Y</math> with a metric <math> d_Y(x,y)</math>. Let <math>c</math> be a limit point of <math>D</math> and let <math>L</math> be a point of <math>Y</math>.
 
We say that
 
: <math> \lim_{x\to c}f(x) = L </math>
 
if for every <math> \varepsilon > 0 </math>, there exists a <math> \delta </math> such that for all <math>x\in D</math>, if <math> 0 < d_X(x,c) < \delta </math>, then <math> d_Y(f(x),L) < \varepsilon</math>.
 
Since <math>d(x,y) = |x-y|</math> is a metric on the real numbers, one can show that this definition generalizes the first definition for real functions.<ref>{{cite book|author=Rudin, Walter|title=Principles of Mathematical Analysis|publisher= McGraw-Hill Science/Engineering/Math |year=1976|isbn= 978-0070542358 |page=[https://archive.org/details/principlesofmath00rudi/page/84 84]|url=https://archive.org/details/principlesofmath00rudi|url-access=registration}}</ref>-->
 
===Negasi dari pernyataan yang tepat===
<!--The [[logical negation]] of the [[definition]] is as follows:<ref>{{cite book|last1=Spivak|first1=Michael|title=Calculus|url=https://archive.org/details/calculus4thediti00mich|url-access=registration|date=2008|publisher=Publish or Perish|location=Houston, Tex.|isbn=978-0914098911|page=[https://archive.org/details/calculus4thediti00mich/page/97 97]|edition=4th}}</ref>
 
Suppose <math>f</math> is defined on a subset <math>D</math> of a metric space <math>X</math> with a metric <math> d_X(x,y)</math> and maps into a metric space <math>Y</math> with a metric <math> d_Y(x,y)</math>. Let <math>c</math> be a limit point of <math>D</math> and let <math>L</math> be a point of <math>Y</math>.
 
We say that
 
: <math> \lim_{x\to c}f(x) \neq L </math>
 
if there exists an <math> \varepsilon > 0 </math> such that for all <math> \delta > 0 </math> there is an <math>x\in D</math> such that <math> 0 < d_X(x,c) < \delta </math> and <math> d_Y(f(x),L) > \varepsilon</math>.
 
We say that <math> \lim_{x\to c}f(x) </math> does not exist if for all <math>L\in Y</math>, <math> \lim_{x\to c}f(x) \neq L </math>.
 
For the negation of a real valued function defined on the real numbers, simply set <math>d_Y(x,y) = d_X(x,y)= |x-y|</math>.-->
 
===Pernyataan yang tepat untuk batas pada tak terhingga===
<!--The precise statement for limits at infinity is as follows:
 
Suppose <math>f</math> is real-valued that is defined on a subset <math>D</math> of the real numbers that contains arbitrarily large values. We say that
 
: <math> \lim_{x\to\infty}f(x) = L </math>
 
if for every <math>\varepsilon > 0</math>, there is a real number <math>N > 0 </math> such that for all <math>x\in D</math>, if <math>x > N</math> then <math>|f(x) - L| < \varepsilon</math>.<ref>{{citation|title = Calculus | edition = 8|first = James |last = Stewart | author-link = James Stewart | publisher = Cengage | year = 2016 | section = Section 3.4}}</ref>
 
It is also possible to give a definition in general metric spaces.{{cn|date=July 2020}}-->
 
==Contoh yang berhasilbekerja==
===Contoh 1===
KamiIni akan tunjukkanmenunjukkan itubahwa
 
: <math>\lim_{x\to 0} x\sin{\left(\frac{1}{x}\right)} = 0
</math>.
 
Kami membiarkanDiberikan <math>\varepsilon > 0</math> be given. Kita perlu menemukan file, <math>\delta >0 </math> sepertidiperlukan yangsehingga <math>|x-0| < \delta </math> menyiratkan <math>\left|x\sin\left(\frac{1}{x}\right) - 0\right| < \varepsilon </math>.
 
Karena [[sinus]] dibatasi di atas <math>1</math> dan di bawahnya oleh −1<math>-1</math>,
 
<math>
Baris 173 ⟶ 90:
</math>
 
Demikianlah, jika kita ambil <math>\delta = \varepsilon</math> dipilih, maka <math>|x| =|x-0| < \delta</math> menyiratkan <math>\left|x\sin{\left(\frac{1}{x}\right)} - 0\right| \leq |x| < \varepsilon </math>, yang melengkapi buktinya.
 
===Contoh 2===
Pernyataan
Mari kita buktikan pernyataan itu
 
: <math> \lim_{x\to a} x^2 = a^2</math>
forakan anydibuktikan realuntuk numbersuatu bilangan real <math>a</math>.
 
mariDiberikan <math>\varepsilon>0</math> diberikan. Kami akan menemukan, <math>\delta > 0 </math> sepertiakan ditemukan yangsehingga <math>|x-a|<\delta</math> menyiratkan <math>|x^2-a^2|<\varepsilon </math>.
 
Kami mulaiDimulai dengan memfaktorkan:
 
: <math> |x^2-a^2| = |(x-a)(x+a)|=|x-a||x+a|.</math>.
 
Kami menyadari ituIstilah <math>|x-a|</math> adalah istilah yang dibatasi oleh <math>\delta</math> jadi kitabatas bisadari mengandaikan1 batasandapat 1kita misalkan, dan kemudian memilih sesuatu yang lebih kecil daridaripadanya dapat diambil ituuntuk <math>\delta</math>.<ref>{{cite book|last1=Spivak|first1=Michael|title=Kalkulus|url=https://archive.org/details/calculus4thediti00mich|url-access=registration|date=2008|publisher=Publish or Perish|location=Houston, Tex.|isbn=978-0914098911|page=[https://archive.org/details/calculus4thediti00mich/page/95 95]|edition=4th}}</ref>
 
Jadi, kamiini kiradianggap bahwa <math> |x-a| < 1 </math>. SetelahKarena <math> |x| - |y| \leq |x-y| </math> berlaku secarapada umumumumnya untuk bilangan real <math>x</math> dan <math>y</math>, kita memiliki
 
: <math> |x| - |a| \leq |x-a| < 1.</math>.
 
Dengan demikian,
 
: <math> |x| < 1 + |a|.</math>.
 
JadiDengan demikian, melalui [[pertidaksamaan segitiga]],
 
: <math> |x+a| \leq |x| + |a| < 2|a| + 1.</math>
 
JadiDengan demikian, jika kita menganggapnya lebih jauh bahwa
 
:<math> |x-a| < \frac{\varepsilon}{2|a| +1}</math>
 
maka
kemudian
 
:<math>|x^2-a^2| <\varepsilon. </math>
 
Singkatnya, <math> \delta = \min{\left(1,\frac{\varepsilon}{2|a| +1}\right)}</math> adalah himpunannya.
Singkatnya, kami menetapkan
: <math> \delta = \min{\left(1,\frac{\varepsilon}{2|a| +1}\right)}.</math>
 
Jadi, jika <math> |x-a|<\delta</math>, setelah itumaka
 
: <math>
Baris 219 ⟶ 135:
&< \frac{\varepsilon}{2|a| +1}(|x+a|)\\
&< \frac{\varepsilon}{2|a| +1}(2|a|+1)\\
&=\varepsilon.
\end{align}
</math>
<!--ThusDengan demikian, wekita havememiliki found asebuah <math>\delta</math> such thatsehingga <math> |x-a| < \delta</math> impliesmenyiratkan <math>|x^2-a^2| <\varepsilon </math>. ThusDengan demikian, wekita havetelah shownmenunjukkan thatbahwa
 
: <math> \lim_{x\to a} x^2 = a^2</math>
for any real number <math>a</math>.-->
 
untuk suatu bilangan real <math>a</math>.
===Contoh 3===
 
<!-- [Bagian ini sedang dalam terjemahan] Let us prove the statement that
=== Contoh 3 ===
Pernyataan
 
: <math>\lim_{x \to 5} (3x - 3) = 12</math>
 
akan dibuktikan.
 
Ini mudah dibuktikan melalui pemahaman grafis limit, dan demikian berfungsi sebagai dasar-dasar yang kuat untuk induksi pembuktiannya. Menurut definisi formal di atas, sebuah pernyataan limit adalah benar jika dan hanya jika membatasi <math>x</math> ke satuan <math>\delta</math> dari <math>c</math> akan pasti membatasi <math>f(x)</math> ke satuan <math>\varepsilon</math> dari <math>L</math>. Dalam kasus yang spesifik, ini berarti bahwa pernyataan tersebut benar jika dan hanya jika membatasi <math>x</math> ke satuan <math>\delta</math> dari 5 akan pasti membatasi
 
: <math>3x - 3</math>
 
ke satuan <math>\varepsilon</math> dari 12. Kunci secara keseluruhan untuk membuktikan implikasi ini adalah untuk menunjukkan bagaimana <math>\delta</math> dan <math>\varepsilon</math> harus berkaitan dengan satu sama lain sehingga implikasinya berlaku. Secara matematis, ini akan menunjukkan bahwa
 
: <math> 0 < | x - 5 | < \delta \ \Rightarrow \ | (3x - 3) - 12 | < \varepsilon </math>.
 
Dengan menyederhanakan, memfaktorkan, dan membagi 3 di ruas kanan implikasi menghasilkan
 
: <math> | x - 5 | < \frac{\varepsilon}{3}</math>,
 
yang secara langsung memberikan nilai yang diperlukan jika
 
: <math> \delta = \varepsilon / 3 </math>
 
dipilih.
 
Dengan demikian, buktinya terselesaikan. Kunci mengenai bukti tersebut terletak dalam kemampuan salah satunya untuk memilih batas-batas di <math>x</math>, dam kemudian menyimpulkan batas-batas berpadanan di <math>f(x)</math>, yang mana dalam kasus ini berkaitan dengan sebuah faktor dari 3, yang secara keseluruhan karena kemiringan dari 3 di garis
: <math>\lim_{x \to 5} (3x - 3) = 12.</math>
 
: <math> y = 3x - 3</math>.
This is easily shown through graphical understandings of the limit, and as such serves as a strong basis for introduction to proof. According to the formal definition above, a limit statement is correct if and only if confining <math>x</math> to <math>\delta</math> units of <math>c</math> will inevitably confine <math>f(x)</math> to <math>\varepsilon</math> units of <math>L</math>. In this specific case, this means that the statement is true if and only if confining <math>x</math> to <math>\delta</math> units of 5 will inevitably confine
 
== Kekontinuan ==
:<math>3x - 3</math>
Sebuah fungsi <math>f</math> dikatakan [[Fungsi kontinu|kontinu]] di <math>c</math> jika keduanya didefinisikan di <math>c</math> dan nilainya di <math>c</math> sama dengan limit dari <math>f</math> ketika <math>x</math> mendekati <math>c</math>:
 
: <math>\lim_{x\to c} f(x) = f(c)</math>.
to <math>\varepsilon</math> units of 12. The overall key to showing this implication is to demonstrate how <math>\delta</math> and <math>\varepsilon</math> must be related to each other such that the implication holds. Mathematically, we want to show that
 
Definisi <math>(\varepsilon, \delta)</math> untuk sebuah fungsi kontinu dapat diperoleh dari definisi limit dengan menggantikan <math>0<|x-c|<\delta</math> dengan <math>|x-c|<\delta</math>, untuk memastikan bahwa <math>f</math> didefinisikan di <math>c</math> dan sama dengan limitnya.
:<math> 0 < | x - 5 | < \delta \ \Rightarrow \ | (3x - 3) - 12 | < \varepsilon . </math>
 
Sebuah fungsi <math>f</math> dikatakan [[Fungsi kontinu|kontinu]] di selang <math>I</math> jiak fungsi <math>f</math> kontinu di setiap titik <math>c</math> dari <math>I</math>.
Simplifying, factoring, and dividing 3 on the right hand side of the implication yields
 
== Perbandingan dengan definisi infinitesimal ==
:<math> | x - 5 | < \varepsilon / 3 ,</math>
[[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>
which immediately gives the required result if we choose
 
== Keluarga definisi limit formal ==
:<math> \delta = \varepsilon / 3 .</math>
Tidak ada definisi limit yang tunggal - adanya seluruh definisi keluarga. Ini dikarenakan kehadiran takhingga, dan konsep limit "dari sebelah kanan"" dan "dari sebelah kiri". Limit itu sendiri dapat menjadi sebuah nilai terhingga, <math>\infty</math>, atau <math>-\infty</math>. Nilai yang mendekati oleh <math>x</math> juga dapat menjadi nilai terhingga, <math>\infty</math>, atau <math>-\infty</math>, dan jika ini merupakan sebuah nilai terhingga, ini dapat mendekati dari kiri atau dari kanan. Biasanya, setiap kombinasinya diberikan definisi itu sendiri, seperti di bawah ini:{{Aligned table|'''Notasi'''|179=<math>{\color{Red}-\infty}</math>|166=<math>{\color{Green}\exists M < 0, }</math>|167=<math>\forall x \in D,</math>|168=|169=<math>x</math>|170=<math>< {\color{Green}M }</math>|171=<math>\Rightarrow</math>|172=<math>{\color{Red}N} <</math>|173=<math>f(x)</math>|174=|175=|176=<math>\lim_{x \to -\infty} x^2 = \infty</math>|177=<math>\lim_{x \to {\color{Green}c}}</math>|178=<math>f(x) =</math>|180=<math>\iff</math>|164=<math>\iff</math>|181=<math>{\color{Red}\forall N < 0,}</math>|182=<math>{\color{Green}\exists \delta > 0,}</math>|183=<math>\forall x \in D,</math>|184=<math>{\color{Green}c-\delta} <</math>|185=<math>x</math>|186=<math>< {\color{Green}c+\delta}</math>|187=<math>\Rightarrow</math>|188=|189=<math>f(x)</math>|190=<math>< {\color{Red}N}</math>|191=|192=<math>\lim_{x \to 0} -|1/x| = -\infty</math>|193=<math>\lim_{x \to {\color{Green}c^+}}</math>|165=<math>{\color{Red}\forall N > 0,}</math>|163=<math>{\color{Red}\infty}</math>|195=<math>{\color{Red}-\infty}</math>|146=<math>f(x) =</math>|133=<math>{\color{Red}\forall N > 0,}</math>|134=<math>{\color{Green}\exists \delta > 0,}</math>|135=<math>\forall x \in D,</math>|136=<math>{\color{Green}c-\delta} <</math>|137=<math>x</math>|138=<math>< {\color{Green}c }</math>|139=<math>\Rightarrow</math>|140=<math>{\color{Red}N} <</math>|141=<math>f(x)</math>|142=|143=|144=<math>\lim_{x \to 0^-} -1/x = \infty</math>|145=<math>\lim_{x \to {\color{Green}\infty}}</math>|147=<math>{\color{Red}\infty}</math>|162=<math>f(x) =</math>|148=<math>\iff</math>|149=<math>{\color{Red}\forall N > 0,}</math>|150=<math>{\color{Green}\exists M > 0, }</math>|151=<math>\forall x \in D,</math>|152=<math>{\color{Green}M } <</math>|153=<math>x</math>|154=|155=<math>\Rightarrow</math>|156=<math>{\color{Red}N} <</math>|157=<math>f(x)</math>|158=|159=|160=<math>\lim_{x \to \infty} e^x = \infty</math>|161=<math>\lim_{x \to {\color{Green}-\infty}}</math>|194=<math>f(x) =</math>|196=<math>\iff</math>|131=<math>{\color{Red}\infty}</math>|244=<math>\iff</math>|231=<math>\forall x \in D,</math>|232=<math>{\color{Green}M } <</math>|233=<math>x</math>|234=|235=<math>\Rightarrow</math>|236=|237=<math>f(x)</math>|238=<math>< {\color{Red}N}</math>|239=|240=<math>\lim_{x \to \infty} -x = -\infty</math>|241=<math>\lim_{x \to {\color{Green}-\infty}}</math>|242=<math>f(x) =</math>|243=<math>{\color{Red}-\infty}</math>|245=<math>{\color{Red}\forall N < 0,}</math>|229=<math>{\color{Red}\forall N < 0,}</math>|246=<math>{\color{Green}\exists M < 0, }</math>|247=<math>\forall x \in D,</math>|248=|249=<math>x</math>|250=<math>< {\color{Green}M }</math>|251=<math>\Rightarrow</math>|252=|253=<math>f(x)</math>|254=<math>< {\color{Red}N}</math>|255=|256=<math>\lim_{x \to -\infty} x^3 = -\infty</math>|cols=16|col8align=right|230=<math>{\color{Green}\exists M > 0, }</math>|228=<math>\iff</math>|197=<math>{\color{Red}\forall N < 0,}</math>|211=<math>{\color{Red}-\infty}</math>|198=<math>{\color{Green}\exists \delta > 0,}</math>|199=<math>\forall x \in D,</math>|200=<math>{\color{Green}c } <</math>|201=<math>x</math>|202=<math>< {\color{Green}c+\delta}</math>|203=<math>\Rightarrow</math>|204=|205=<math>f(x)</math>|206=<math>< {\color{Red}N}</math>|207=|208=<math>\lim_{x \to 0^+} \log(x) = -\infty</math>|209=<math>\lim_{x \to {\color{Green}c^-}}</math>|210=<math>f(x) =</math>|212=<math>\iff</math>|227=<math>{\color{Red}-\infty}</math>|213=<math>{\color{Red}\forall N < 0,}</math>|214=<math>{\color{Green}\exists \delta > 0,}</math>|215=<math>\forall x \in D,</math>|216=<math>{\color{Green}c-\delta} <</math>|217=<math>x</math>|218=<math>< {\color{Green}c }</math>|219=<math>\Rightarrow</math>|220=|221=<math>f(x)</math>|222=<math>< {\color{Red}N}</math>|223=|224=<math>\lim_{x \to 0^-} 1/x = -\infty</math>|225=<math>\lim_{x \to {\color{Green}\infty}}</math>|226=<math>f(x) =</math>|132=<math>\iff</math>|130=<math>f(x) =</math>||49=<math>\lim_{x \to {\color{Green}c^-}}</math>|36=<math>\iff</math>|37=<math>{\color{Red}\forall\varepsilon > 0,}</math>|38=<math>{\color{Green}\exists \delta > 0,}</math>|39=<math>\forall x \in D,</math>|40=<math>{\color{Green}c } <</math>|41=<math>x</math>|42=<math>< {\color{Green}c+\delta}</math>|43=<math>\Rightarrow</math>|44=<math>{\color{Red}L-\varepsilon} <</math>|45=<math>f(x)</math>|46=<math>< {\color{Red}L+\varepsilon}</math>|47=|48=<math>\lim_{x \to 0^+} x^2 + \sgn(x) = 1</math>|50=<math>f(x) =</math>|34=<math>f(x) =</math>|51=<math>{\color{Red}L}</math>|52=<math>\iff</math>|53=<math>{\color{Red}\forall\varepsilon > 0,}</math>|54=<math>{\color{Green}\exists \delta > 0,}</math>|55=<math>\forall x \in D,</math>|56=<math>{\color{Green}c-\delta} <</math>|57=<math>x</math>|58=<math>< {\color{Green}c }</math>|59=<math>\Rightarrow</math>|60=<math>{\color{Red}L-\varepsilon} <</math>|61=<math>f(x)</math>|62=<math>< {\color{Red}L+\varepsilon}</math>|63=|35=<math>{\color{Red}L}</math>|33=<math>\lim_{x \to {\color{Green}c^+}}</math>|65=<math>\lim_{x \to {\color{Green}\infty}}</math>|16='''Contoh'''|||'''Definisi'''|||||||||||17=<math>\lim_{x \to {\color{Green}c}}</math>|32=<math>\lim_{x \to 0} \sin(x) = 0</math>|18=<math>f(x) =</math>|19=<math>{\color{Red}L}</math>|20=<math>\iff</math>|21=<math>{\color{Red}\forall\varepsilon > 0,}</math>|22=<math>{\color{Green}\exists \delta > 0,}</math>|23=<math>\forall x \in D,</math>|24=<math>{\color{Green}c-\delta} <</math>|25=<math>x</math>|26=<math>< {\color{Green}c+\delta}</math>|27=<math>\Rightarrow</math>|28=<math>{\color{Red}L-\varepsilon} <</math>|29=<math>f(x)</math>|30=<math>< {\color{Red}L+\varepsilon}</math>|31=|64=<math>\lim_{x \to 0^-} x^2 + \sgn(x) = -1</math>|66=<math>f(x) =</math>|129=<math>\lim_{x \to {\color{Green}c^-}}</math>|114=<math>f(x) =</math>|101=<math>{\color{Red}\forall N > 0,}</math>|102=<math>{\color{Green}\exists \delta > 0,}</math>|103=<math>\forall x \in D,</math>|104=<math>{\color{Green}c-\delta} <</math>|105=<math>x</math>|106=<math>< {\color{Green}c+\delta}</math>|107=<math>\Rightarrow</math>|108=<math>{\color{Red}N} <</math>|109=<math>f(x)</math>|110=|111=|112=<math>\lim_{x \to 0} |1/x| = \infty</math>|113=<math>\lim_{x \to {\color{Green}c^+}}</math>|115=<math>{\color{Red}\infty}</math>|99=<math>{\color{Red}\infty}</math>|116=<math>\iff</math>|117=<math>{\color{Red}\forall N > 0,}</math>|118=<math>{\color{Green}\exists \delta > 0,}</math>|119=<math>\forall x \in D,</math>|120=<math>{\color{Green}c } <</math>|121=<math>x</math>|122=<math>< {\color{Green}c+\delta}</math>|123=<math>\Rightarrow</math>|124=<math>{\color{Red}N} <</math>|125=<math>f(x)</math>|126=|127=|128=<math>\lim_{x \to 0^+} 1/x = \infty</math>|100=<math>\iff</math>|98=<math>f(x) =</math>|67=<math>{\color{Red}L}</math>|81=<math>\lim_{x \to {\color{Green}-\infty}}</math>|68=<math>\iff</math>|69=<math>{\color{Red}\forall\varepsilon > 0,}</math>|70=<math>{\color{Green}\exists M > 0, }</math>|71=<math>\forall x \in D,</math>|72=<math>{\color{Green}M } <</math>|73=<math>x</math>|74=|75=<math>\Rightarrow</math>|76=<math>{\color{Red}L-\varepsilon} <</math>|77=<math>f(x)</math>|78=<math>< {\color{Red}L+\varepsilon}</math>|79=|80=<math>\lim_{x \to \infty} 1/x = 0</math>|82=<math>f(x) =</math>|97=<math>\lim_{x \to {\color{Green}c}}</math>|83=<math>{\color{Red}L}</math>|84=<math>\iff</math>|85=<math>{\color{Red}\forall\varepsilon > 0,}</math>|86=<math>{\color{Green}\exists M < 0, }</math>|87=<math>\forall x \in D,</math>|88=|89=<math>x</math>|90=<math>< {\color{Green}M }</math>|91=<math>\Rightarrow</math>|92=<math>{\color{Red}L-\varepsilon} <</math>|93=<math>f(x)</math>|94=<math>< {\color{Red}L+\varepsilon}</math>|95=|96=<math>\lim_{x \to -\infty} e^x = 0</math>|col12align=right}}
 
== Lihat pula ==
Thus the proof is completed. The key to the proof lies in the ability of one to choose boundaries in <math>x</math>, and then conclude corresponding boundaries in <math>f(x)</math>, which in this case were related by a factor of 3, which is entirely due to the slope of 3 in the line
 
* [[Fungsi kontinu]]
:<math> y = 3x - 3 .</math>-->
* [[Limit barisan]]
* [[Daftar topik kalkulus]]
* [[Teorema apit]]
 
== Referensi ==
Diperoleh dari "https://wiki-indonesia.club/wiki/Definisi_limit_(ε,_δ)"