05626265

设\[\mathop {\lim }\limits_{x \to \begin{array}{*{20}{c}}
{{a^ + }}
\end{array}} f\left( x \right) = \mathop {\lim }\limits_{x \to \begin{array}{*{20}{c}}
{ + \infty }
\end{array}} f\left( x \right) = A\]

其中$A$是有限数或$\pm \infty $

若$f\left( x \right) = A$,则结论显然成立；若$f\left( x \right) \ne A$,则存在${x_0} \in \left( {a, + \infty } \right)$,使得$f\left( {{x_0}} \right) \ne A$.

不妨设$f\left( {{x_0}} \right) > A$,则由实数的稠密性知,存在${\varepsilon _0} > 0$,使得\[f\left( {{x_0}} \right) > f\left( {{x_0}} \right) - {\varepsilon _0} > A\]

由$\lim \limits_{x \to \begin{array}{*{20}{c}}
{{a^ + }}
\end{array}} f\left( x \right) = A < A + {\varepsilon _0}$及极限的保号性知
\[\exists \delta > 0,\forall x \in \left( {a,a + \delta } \right),有f\left( x \right) < A + {\varepsilon _0}\]

特别地,取${x_1} \in \left( {a,a + \delta } \right)$,且${x_1} < {x_0}$,则
\[f\left( {{x_1}} \right) < A + {\varepsilon _0} < f\left( {{x_0}} \right)\]

由连续函数介值定理知,存在${\xi _1} \in \left( {{x_1},{x_0}} \right)$,使得
\[f\left( {{\xi _1}} \right) = A + {\varepsilon _0}\]

由$\lim \limits_{x \to \begin{array}{*{20}{c}}
{ + \infty }
\end{array}} f\left( x \right) = A < A + {\varepsilon _0}$及极限的保号性知
\[\exists M > a,\forall x > M,有f\left( x \right) < A + {\varepsilon _0}\]

特别地,取${x_2} \in \left( {M, + \infty } \right)$,且${x_0} < {x_2}$,则
\[f\left( {{x_2}} \right) < A + {\varepsilon _0} < f\left( {{x_0}} \right)\]

由连续函数介值定理知,存在${\xi _2} \in \left( {{x_0},{x_2}} \right)$,使得
\[f\left( {{\xi _2}} \right) = A + {\varepsilon _0}\]

由$Rolle$中值定理知,存在$\xi \in \left( {{\xi _1},{\xi _2}} \right)$,使得
\[f'\left( \xi \right) = 0\]