Source / pinned edition
The Precise Definition of a Limit
Source of package openstax.calculus_limits · OpenStax Calculus, ch. 2 Limits, module m53493
- work
- urn:openstax:clir:calculus-limits#M53493
- number
- OpenStax Calculus, ch. 2 Limits, module m53493
- kind
- urn:openstax:clir:calculus-limits#treatise
m53493/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
- edition
- urn:openstax:clir:calculus-limits#M53493_MAIN
- language
- en
- officiality
- official
- materialisation
- PINNED_OFFICIAL_BYTES
- life cycle
- adopted: 2026-07-15
Publications
| URI | Type | documentHash | Retrieved |
|---|---|---|---|
| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53493/index.cnxml | application/xml; charset=utf-8 | sha256:e5e046f267dee46ca85a9cfc76847ff31ebb71a9205b5961e1152f2ea77bc666 | 2026-09-12T00:00:00Z |
Fragments — 1
module/m53493#fs-id1170572641433 · defn · referenced by: 3
en · official · sha256:3193e5de60c09b256f5e97ed07a9c5d602b907bc4285ca833ebb47ee3e2f0fe4
<note id="fs-id1170572641433">
<title>Definition</title>
<para id="fs-id1170572482431">Let <m:math><m:mrow><m:mi>f</m:mi><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:math> be defined for all <m:math><m:mrow><m:mi>x</m:mi><m:mo>≠</m:mo><m:mi>a</m:mi></m:mrow></m:math> over an open interval containing <emphasis effect="italics">a</emphasis>. Let <emphasis effect="italics">L</emphasis> be a real number. Then</para>
<equation id="fs-id1170571542392" class="unnumbered"><label/><m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>x</m:mi><m:mo stretchy="false">→</m:mo><m:mi>a</m:mi></m:mrow></m:munder><m:mi>f</m:mi><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo><m:mo>=</m:mo><m:mi>L</m:mi></m:mrow></m:math></equation>
<para id="fs-id1170572217764">if, for every <m:math><m:mrow><m:mi>ε</m:mi><m:mo>></m:mo><m:mn>0</m:mn><m:mo>,</m:mo></m:mrow></m:math> there exists a <m:math><m:mrow><m:mi>δ</m:mi><m:mo>></m:mo><m:mn>0</m:mn><m:mo>,</m:mo></m:mrow></m:math> such that if <m:math><m:mrow><m:mn>0</m:mn><m:mo><</m:mo><m:mo>|</m:mo><m:mi>x</m:mi><m:mo>−</m:mo><m:mi>a</m:mi><m:mo>|</m:mo><m:mo><</m:mo><m:mi>δ</m:mi><m:mo>,</m:mo></m:mrow></m:math> then <m:math><m:mrow><m:mo>|</m:mo><m:mi>f</m:mi><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo><m:mo>−</m:mo><m:mi>L</m:mi><m:mo>|</m:mo><m:mo><</m:mo><m:mi>ε</m:mi><m:mo>.</m:mo></m:mrow></m:math></para>
</note>Referencing norms: epsilon_delta, epsilon_delta/necessary, epsilon_delta_certificate
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