21 lines
678 B
Markdown
21 lines
678 B
Markdown
---
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up: "[[dérivation]]"
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tags: "#s/maths/analyse"
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---
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> [!proposition]+
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> Soit $x_0 \in E$ un point
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> Soient $f, g \in \mathcal{D}^{1}(E, F)$ deux fonctions dérivables avec $f(x_0)= g(x_0) = 0$
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> $$\lim_{x\rightarrow x_0} \dfrac{f(x)}{g(x)} = \dfrac{f'(x_0)}{g'(x_0)}$$
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> - ! Il faut que $f(x_0) = 0$ et $g(x_0) = 0$
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^theoreme
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> [!démonstration] Démonstration
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> $$\begin{aligned}
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> \lim_{x\rightarrow x_0} \dfrac{f(x)}{g(x)} &= \dfrac{f'(x)}{g'(x)}\\[3ex]
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> &= \lim_{x\rightarrow x_0} \dfrac{\dfrac{f(x)}{x-x_0}}{\dfrac{g(x)}{x-x_0}}\\[3ex]
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> &= \lim_{x\rightarrow x_0} \dfrac{\dfrac{f(x) - f(x_0)}{x-x_0}}{\dfrac{g(x)-g(x_0)}{x-x_0}}\\
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> \end{aligned}$$
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^demonstration
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