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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 [[real number]]s by [[Richard Dedekind]].<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|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|title=Structure and randomness : pages from year one of a mathematical blog|date=2008|publisher=American Mathematical Society|location=Providence, R.I.|isbn=978-0-8218-4695-7|pages=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==
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