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{{Short description|Definisi matematis limit}}{{Under construction}}{{DISPLAYTITLE:(''ε'', ''δ'')-definisi limit}}
{{DISPLAYTITLE:(''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'']]
 
Dalam [[kalkulus]], '''(''ε'',&nbsp;''δ'')-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 (<math>\varepsilon,\delta</math>) -definisi bataslimit 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 38 ⟶ 37:
|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}}▼
 
▲AdditionallySebagai tambahan, Newton occasionallyterkadang explainedmenjelaskan limitslimit indalam termsistilah similaryang toserupa thedengan epsilon–deltadefinisi definitionepsilon–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]] developedmengembangkan ansebuah infinitesimal ofoleh hisdirinya owndan andmencoba trieduntuk tomemberikannya providedengan itsebuah withdasar ayang rigorous footingsetepat-tepatnya, buttetapi itini wastetap stilldisambut greeteddengan withgelisah uneaseoleh bybeberapa somematematikawan mathematiciansdan andpara philosophersfilsafat.<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|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]] gavememberikan asebuah definition ofdefinisi limit indalam termshal ofgagasan alebih moreprimitif primitiveyang notiondisebut he called asebuah ''variablekuantitas quantityvariabel''. HeDia nevertidak gavepernah anmemberikan epsilon–delta definition ofdefinisi limit (Grabiner 1981). SomeBeberapa ofbukti Cauchy's proofsberisi containindikasi indications of themetode 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}}
:<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|urllocation=https://archiveProvidence, R.I.org/details/structurerandomn00taot|date=2008|publisher=American Mathematical Society|location=Providence, R.I.|isbn=978-0-8218-4695-7|pages=[https://archive.org/details/structurerandomn00taot/page/95 95]–110}}</ref>-->
 
==Contoh yang bekerja==
===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 104 ⟶ 89:
</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 150 ⟶ 134:
&< \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>
 
foruntuk anysuatu bilangan real number <math>a</math>.-->
 
== Kekontinuan ==
 
== Perbandingan dengan definisi infinitesimal ==
 
== Keluarga definisi limit formal ==
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|'''Notation'''|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='''Example'''|||'''Def.'''|||||||||||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}}
 
== Referensi ==
Diperoleh dari "https://wiki-indonesia.club/wiki/Definisi_limit_(ε,_δ)"