Discussão:Regra de l'Hôpital

Addition to the article

Addition to the article "Regra de l'Hopital"

$~t\vdash \quad \mathrm {f} :\mathbb {X} _{\mathrm {f} }\mapsto \mathbb {Y} _{\mathrm {f} }\ \ \land \ \ \mathbb {X} _{\mathrm {f} }\times \mathbb {Y} _{\mathrm {f} }\subseteq \mathbb {R} ^{2}\quad \land \quad \mathrm {g} :\mathbb {X} _{\mathrm {g} }\mapsto \mathbb {Y} _{\mathrm {g} }\ \ \land \ \ \mathbb {X} _{\mathrm {g} }\times \mathbb {Y} _{\mathrm {g} }\subseteq \mathbb {R} ^{2}\quad \land \quad a\in \mathbb {X} _{\mathrm {f} }\ \cap \ \mathbb {X} _{\mathrm {g} }\quad \land$

$~\lim _{x\to a}f(x)=\lim _{x\to a}g(x)\quad \land \quad \{\lim _{x\to a}f(x),\ \ \lim _{x\to a}g(x)\}\ \subseteq \ \{\{-\infty \},\ \{0\},\ \{\infty \}\}\quad \land$

$~\exists _{\delta \ \in \ (0,\ \infty )}\ \forall _{x\ \in \ \mathbb {X} _{\mathrm {f} }\ \cap \ \mathbb {X} _{\mathrm {g} }\ \land \ 0\ <\ |x-a|\ <\ \delta }\ (f'(x)\in \mathbb {R} \quad \land \quad g'(x)\in \mathbb {R} \setminus \{0\})\quad \to$

$~(\exists _{L\ \in \ \mathbb {R} }\ (\lim _{x\to a}{\frac {f'(x)}{g'(x)}}=L)\quad \to \quad \lim _{x\to a}{\frac {f(x)}{g(x)}}=\lim _{x\to a}{\frac {f'(x)}{g'(x)}}\ )$

Notes

$~\mathrm {f} :\mathbb {X} _{\mathrm {f} }\mapsto \mathbb {Y} _{\mathrm {f} }\quad \Leftrightarrow \quad \mathrm {f} =\{\langle x,y\rangle |\quad \langle x,y\rangle \in \mathbb {X} _{\mathrm {f} }\times \mathbb {Y} _{\mathrm {f} }\quad \land \quad y=f(x)\ \}$

$~\vdash \quad \mathbb {X} _{\mathrm {f} }\times \mathbb {Y} _{\mathrm {f} }\ =\ \{\langle x,y\rangle |\quad x\in \mathbb {X} _{\mathrm {f} }\quad \land \quad y\in \mathbb {Y} _{\mathrm {f} }\ \}$

$~\mathbb {X} _{\mathrm {f} }\times \mathbb {Y} _{\mathrm {f} }\subseteq \mathbb {R} ^{2}\quad \Leftrightarrow \quad \forall x\forall y\ (\ \langle x,y\rangle \in \mathbb {X} _{\mathrm {f} }\times \mathbb {Y} _{\mathrm {f} }\ \to \ \langle x,y\rangle \in \mathbb {R} ^{2})$

$~\mathrm {g} :\mathbb {X} _{\mathrm {g} }\mapsto \mathbb {Y} _{\mathrm {g} }\quad \Leftrightarrow \quad \mathrm {g} =\{\langle x,y\rangle |\quad \langle x,y\rangle \in \mathbb {X} _{\mathrm {g} }\times \mathbb {Y} _{\mathrm {g} }\quad \land \quad y=g(x)\ \}$

$~\vdash \quad \mathbb {X} _{\mathrm {g} }\times \mathbb {Y} _{\mathrm {g} }\ =\ \{\langle x,y\rangle |\quad x\in \mathbb {X} _{\mathrm {g} }\quad \land \quad y\in \mathbb {Y} _{\mathrm {g} }\ \}$

$~\mathbb {X} _{\mathrm {g} }\times \mathbb {Y} _{\mathrm {g} }\subseteq \mathbb {R} ^{2}\quad \Leftrightarrow \quad \forall x\forall y\ (\ \langle x,y\rangle \in \mathbb {X} _{\mathrm {g} }\times \mathbb {Y} _{\mathrm {g} }\ \to \ \langle x,y\rangle \in \mathbb {R} ^{2})$

$~a\in \mathbb {X} _{\mathrm {f} }\cap \mathbb {X} _{\mathrm {g} }\quad \Leftrightarrow \quad a\in \mathbb {X} _{\mathrm {f} }\ \ \land \ \ a\in \mathbb {X} _{\mathrm {g} }$

$~\vdash \quad \mathbb {X} _{\mathrm {f} }\cap \mathbb {X} _{\mathrm {g} }\ =\ \{x|\quad x\in \mathbb {X} _{\mathrm {f} }\quad \land \quad x\in \mathbb {X} _{\mathrm {g} }\ \}$

$~\lim _{x\to a}f(x)=-\infty \quad \Leftrightarrow \quad \forall _{m\ \in \ (-\infty ,\ 0)}\ \exists _{\delta \ \in \ (0,\ \infty )}\ \forall _{x\ \in \ \mathbb {X} _{\mathrm {f} }}\ (0<|x-a|<\delta \ \to \ f(x)<m)$

$~\lim _{x\to a}g(x)=-\infty \quad \Leftrightarrow \quad \forall _{m\ \in \ (-\infty ,\ 0)}\ \exists _{\delta \ \in \ (0,\ \infty )}\ \forall _{x\ \in \ \mathbb {X} _{\mathrm {g} }}\ (0<|x-a|<\delta \ \to \ g(x)<m)$

$~\lim _{x\to a}f(x)=0\quad \Leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\ \infty )}\ \exists _{\delta \ \in \ (0,\ \infty )}\ \forall _{x\ \in \ \mathbb {X} _{\mathrm {f} }}\ (0<|x-a|<\delta \ \to \ |f(x)|<\varepsilon )$

$~\lim _{x\to a}g(x)=0\quad \Leftrightarrow \quad \forall _{\varepsilon \ \in \ (0,\ \infty )}\ \exists _{\delta \ \in \ (0,\ \infty )}\ \forall _{x\ \in \ \mathbb {X} _{\mathrm {g} }}\ (0<|x-a|<\delta \ \to \ |g(x)|<\varepsilon )$

$~\lim _{x\to a}f(x)=\infty \quad \Leftrightarrow \quad \forall _{M\ \in \ (0,\ \infty )}\ \exists _{\delta \ \in \ (0,\ \infty )}\ \forall _{x\ \in \ \mathbb {X} _{\mathrm {f} }}\ (0<|x-a|<\delta \ \to \ f(x)>M)$

$~\lim _{x\to a}g(x)=\infty \quad \Leftrightarrow \quad \forall _{M\ \in \ (0,\ \infty )}\ \exists _{\delta \ \in \ (0,\ \infty )}\ \forall _{x\ \in \ \mathbb {X} _{\mathrm {g} }}\ (0<|x-a|<\delta \ \to \ g(x)>M)$

$~\vdash \quad f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

$~\vdash \quad g'(x)=\lim _{h\to 0}{\frac {g(x+h)-g(x)}{h}}$

$~g'(x)\in \mathbb {R} \setminus \{0\}\quad \Leftrightarrow \quad g'(x)\in \mathbb {R} \ \ \land \ \ g'(x)\notin \{0\}\quad \Leftrightarrow \quad g'(x)\in \mathbb {R} \ \ \land \ \ g'(x)\neq 0$

$~\exists _{L\ \in \ \mathbb {R} }\ (\lim _{x\to a}{\frac {f'(x)}{g'(x)}}=L)\quad \Leftrightarrow \quad \exists L\ (L\in \mathbb {R} \ \ \land \ \ \lim _{x\to a}{\frac {f'(x)}{g'(x)}}=L)\quad \Leftrightarrow \quad \lim _{x\to a}{\frac {f'(x)}{g'(x)}}\in \mathbb {R}$

Cauchy

@João: Teria alguma referência para esta afirmação que inseriu em 2012? Ela foi removida recentemente, mas como não tinha fontes, não quis desfazer. Helder 17h45min de 19 de fevereiro de 2019 (UTC)