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id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div style="margin:1.5em; border:5px solid blue; padding: 1em; background-color:#DDEFD6; font-align:font-size:100%;font-size: 1.4em"><span style=""> L'Hôpital's rule</span></div> <div style="margin:1.5em; border:5px solid blue; padding: 1em; background-color:#DDEFD6; font-align:font-size:100%;font-size: 1.4em"> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}I\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>I</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}I\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8edf5906827fc8f86b0219b96427715bd55a7ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.948ex; height:2.343ex;" alt="{\displaystyle {}I\subseteq \mathbb {R} }"></span> denote an <a href="/wiki/Real_numbers/Intervals/Definition" title="Real numbers/Intervals/Definition">open interval</a>, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}a\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>a</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}a\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/851f3e08e6e3842d9c9d641bc7f9b7c587720053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.242ex; height:2.176ex;" alt="{\displaystyle {}a\in I}"></span> denote a point. Suppose that </p> <dl><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon I\longrightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>I</mi> <mo stretchy="false">⟶</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon I\longrightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c648a5eb286844261f91e6d6bcbbd5e934f073b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.408ex; height:2.509ex;" alt="{\displaystyle f,g\colon I\longrightarrow \mathbb {R} }"></span> </dd></dl> <p>are <a href="/wiki/Real_function/Continuity_in_a_Point/General/Definition" title="Real function/Continuity in a Point/General/Definition">continuous functions</a>, which are <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}I\setminus \{a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>I</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}I\setminus \{a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1751806a1d67ab3e8937efc37ec88feffd6064ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.921ex; height:2.843ex;" alt="{\displaystyle {}I\setminus \{a\}}"></span>, fulfilling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}f(a)=g(a)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}f(a)=g(a)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff320227b8134461488700bd586da2c9c58ba15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.832ex; height:2.843ex;" alt="{\displaystyle {}f(a)=g(a)=0}"></span>, and with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}g'(x)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}g'(x)\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/057a0a3580d030f09133a89661fa8398c3d1cfa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.203ex; height:3.009ex;" alt="{\displaystyle {}g'(x)\neq 0}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}x\neq a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>x</mi> <mo>≠</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}x\neq a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d16e5af18e94a3b332f577a27f4d90c80518cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.658ex; height:2.676ex;" alt="{\displaystyle {}x\neq a}"></span>. Moreover, suppose that the <a href="/wiki/Function/R/Limit/Epsilon/Definition" title="Function/R/Limit/Epsilon/Definition">limit</a> </p> <dl><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}w:=\operatorname {lim} _{x\rightarrow a}\,{\frac {f'(x)}{g'(x)}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>w</mi> <mo>:=</mo> <msub> <mi>lim</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}w:=\operatorname {lim} _{x\rightarrow a}\,{\frac {f'(x)}{g'(x)}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d126afd502f0a52ddcbc4233a8e1d1b749a537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.466ex; height:6.509ex;" alt="{\displaystyle {}w:=\operatorname {lim} _{x\rightarrow a}\,{\frac {f'(x)}{g'(x)}}\,}"></span></dd></dl> <p>exists. </p> <div style="font-weight:bold"> Then also the limit <dl><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {lim} _{x\rightarrow a}\,{\frac {f(x)}{g(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>lim</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {lim} _{x\rightarrow a}\,{\frac {f(x)}{g(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db206ae9fed113f8d5d0d310753caca1f2d7eb08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.943ex; height:6.509ex;" alt="{\displaystyle \operatorname {lim} _{x\rightarrow a}\,{\frac {f(x)}{g(x)}}}"></span> </dd></dl> exists, and it also equals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a081b959224a8003fbaa26385f8c372743878767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle {}w}"></span>. </div> </div> <div class="center"><a href="/wiki/Hospital/Differentiable_in_inner_interval/Fact/Proof" title="Hospital/Differentiable in inner interval/Fact/Proof">Proof</a>, <a class="external text" href="https://en.wikiversity.org/w/index.php?title=Hospital/Differentiable_in_inner_interval/Fact/Proof2&action=edit">Write another proof</a> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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