Источник / закреплённая редакция
L’Hôpital’s Rule
Источник пакета openstax.calculus_limits · OpenStax Calculus, ch. 4 Applications of Derivatives, module m53619
- work
- urn:openstax:clir:calculus-limits#M53619
- номер
- OpenStax Calculus, ch. 4 Applications of Derivatives, module m53619
- вид
- urn:openstax:clir:calculus-limits#treatise
m53619/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
- edition
- urn:openstax:clir:calculus-limits#M53619_MAIN
- язык
- en
- официальность
- official
- материализация
- PINNED_OFFICIAL_BYTES
- жизненный цикл
- adopted: 2026-07-15
Публикации
| URI | Тип | documentHash | Получено |
|---|---|---|---|
| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53619/index.cnxml | application/xml; charset=utf-8 | sha256:47a9cdb551f68cdd646df68ff7da9b9226fcd1d896af5e203ceb85d170ef9ce4 | 2026-09-12T00:00:00Z |
Фрагменты — 2
module/m53619#fs-id1165042330826 · section · на него ссылаются: 1
en · official · sha256:204c9997191b9ede4863c77cc8c11b0b7a3bc3275e4a56228db6b3f68df4126aEN
<note id="fs-id1165042330826" class="theorem">
<title>L’Hôpital’s Rule <m:math><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>∞</m:mi><m:mtext>/</m:mtext><m:mi>∞</m:mi></m:mrow></m:math> Case)</title>
<para id="fs-id1165043426174">Suppose <m:math><m:mi>f</m:mi></m:math> and <m:math><m:mi>g</m:mi></m:math> are differentiable functions over an open interval containing <m:math><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo></m:mrow></m:math> except possibly at <m:math><m:mrow><m:mi>a</m:mi><m:mo>.</m:mo></m:mrow></m:math> Suppose <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:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>∞</m:mi></m:mrow></m:math> (or <m:math><m:mrow><m:mtext>−</m:mtext><m:mi>∞</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:math> and <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>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>∞</m:mi></m:mrow></m:math> (or <m:math><m:mrow><m:mtext>−</m:mtext><m:mi>∞</m:mi><m:mo stretchy="false">)</m:mo><m:mo>.</m:mo></m:mrow></m:math> Then,</para>
<equation id="fs-id1165042373703" 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:mfrac><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mrow><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:mfrac><m:mo>=</m:mo><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:mfrac><m:mrow><m:msup><m:mi>f</m:mi><m:mo>′</m:mo></m:msup><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mrow><m:msup><m:mi>g</m:mi><m:mo>′</m:mo></m:msup><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:mfrac><m:mo>,</m:mo></m:mrow></m:math></equation>
<para id="fs-id1165042707280">assuming the limit on the right exists or is <m:math><m:mi>∞</m:mi></m:math> or <m:math><m:mrow><m:mtext>−</m:mtext><m:mi>∞</m:mi><m:mo>.</m:mo></m:mrow></m:math> This result also holds if the limit is infinite, if <m:math><m:mrow><m:mi>a</m:mi><m:mo>=</m:mo><m:mi>∞</m:mi></m:mrow></m:math> or <m:math><m:mrow><m:mtext>−</m:mtext><m:mi>∞</m:mi><m:mo>,</m:mo></m:mrow></m:math> or the limit is one-sided.</para>
</note>Ссылаются нормы: JustifyLhopital19
module/m53619#fs-id1165043352593 · section · на него ссылаются: 1
en · official · sha256:85d6fab2cecf7e41a3b73a76bee847330d7b6aaaa810882abddc648c5271890cEN
<note id="fs-id1165043352593" class="theorem">
<title>L’Hôpital’s Rule (0/0 Case)</title>
<para id="fs-id1165043276140">Suppose <m:math><m:mi>f</m:mi></m:math> and <m:math><m:mi>g</m:mi></m:math> are differentiable functions over an open interval containing <m:math><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo></m:mrow></m:math> except possibly at <m:math><m:mrow><m:mi>a</m:mi><m:mo>.</m:mo></m:mrow></m:math> If <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:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow></m:math> and <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>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mn>0</m:mn><m:mo>,</m:mo></m:mrow></m:math> then</para>
<equation id="fs-id1165043020148" 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:mfrac><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mrow><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:mfrac><m:mo>=</m:mo><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:mfrac><m:mrow><m:msup><m:mi>f</m:mi><m:mo>′</m:mo></m:msup><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mrow><m:msup><m:mi>g</m:mi><m:mo>′</m:mo></m:msup><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:mfrac><m:mo>,</m:mo></m:mrow></m:math></equation>
<para id="fs-id1165043035673">assuming the limit on the right exists or is <m:math><m:mi>∞</m:mi></m:math> or <m:math><m:mrow><m:mtext>−</m:mtext><m:mi>∞</m:mi><m:mo>.</m:mo></m:mrow></m:math> This result also holds if we are considering one-sided limits, or if <m:math><m:mrow><m:mi>a</m:mi><m:mo>=</m:mo><m:mi>∞</m:mi><m:mspace width="0.2em"/><m:mtext>and</m:mtext><m:mspace width="0.2em"/><m:mo>−</m:mo><m:mi>∞</m:mi><m:mo>.</m:mo></m:mrow></m:math></para>
</note>Ссылаются нормы: JustifyLhopital18
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