胜地不常，盛筵难再，兰亭已矣，梓泽丘墟———《滕王阁序》
(L’Hospital law) Suppose \(f\colon (a,b)\rightarrow \mathbb R\) and \(g\colon(a,b)\rightarrow \mathbb R\) are differiential in \((a,b)\) (\(-\infty\le a<b\le +\infty\)). \(g'(x)\ne 0\) in \((a,b)\) and
\[\dfrac{f(x)}{g(x)}\rightarrow A,\ \ x\rightarrow a^{+}\ (-\infty\le A\le +\infty). \]When either (1) or (2) given by
(1) \(f(x)\rightarrow 0\) and \(g(x)\rightarrow 0\) as \(x\rightarrow a^{+}\).
(2) \(g(x)\rightarrow \infty\) as \(x\rightarrow a^{+}\).
happens, we have
\[\dfrac{f(x)}{g(x)}\rightarrow A,\ \ x\rightarrow a^{+}. \]We can rewrite this result by replacing \(x\rightarrow a^{+}\) by \(x\rightarrow b^{-}\).
Proof: Since \(g'(x)\ne 0\), \(g(x)\) is strictly monotonous in \((a,b)\). Thus when either (1) or (2) happens, there exists \(\delta>0\) such that \(g(x)\ne 0\) in \((a,a+\delta)\). Now we replace \(b\) by \(a+\delta.\)
Fix \(x\) in \((a,b)\). For any \(y\) in \((a,b)\), due to Cauchy's mean value theorem, there exists \(\xi\) between \(x\) and \(y\) such that
\[\dfrac{f(x)-f(y)}{g(x)-g(y)}=\dfrac{f'(\xi)}{g'(\xi)}. \]We rewrite this as
\[\dfrac{f(x)}{g(x)}=\dfrac{f(y)}{g(x)}+\dfrac{f'(\xi)}{g'(\xi)}\left(1-\dfrac{g(y)}{g(x)}\right). \]If (1), there exists \(\delta_1>0,\delta_2>0\) such that
\[\begin{aligned}y\in (a,a+\delta_1)&\implies |f(y)|<|g(x)|(x-a),\\y\in (a,a+\delta_2)&\implies |g(y)|<|g(x)|(x-a).\end{aligned} \]Pick \(y_x\in (a,a+\min\{\delta_1,\delta_2,x-a\}).\)
If (2), there exists \(\delta_3>0\) such that
\[y\in (a,a+\delta_3)\implies |g(x)|>|g(y)|/(x-a) \]Pick \(y_x\in (a,a+\min\{\delta_3,x-a\})\).
Hence \(\dfrac{f(y_x)}{g(x)},\dfrac{g(y_x)}{g(x)}\rightarrow 0\) as \(x\rightarrow a^{+}\).
Thus there exists \(\xi_x\in (y_x,x)\) such that
\[\dfrac{f(x)}{g(x)}=\dfrac{f(y_x)}{g(x)}+\dfrac{f'(\xi_x)}{g'(\xi_x)}\left(1-\dfrac{g(y_x)}{g(x)}\right). \]Obviously \(y_x,\xi_x\rightarrow a^{+}\) as \(x\rightarrow a^{+}\), hence
\[\lim_{x\rightarrow a^{+}}\dfrac{f(x)}{g(x)}=\lim_{x\rightarrow a^{+}}\dfrac{f(y_x)}{g(x)}+\lim_{x\rightarrow a^{+}}\dfrac{f'(\xi_x)}{g'(\xi_x)}\left(1-\lim_{x\rightarrow a^{+}}\dfrac{g(y_x)}{g(x)}\right)=\lim_{\xi\rightarrow a^{+}}\dfrac{f'(\xi)}{g'(\xi)}=A. \] 标签：洛必达,xi,法则,exists,dfrac,there,形式,delta,rightarrow From： https://www.cnblogs.com/space-of-mistery/p/18544725