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时间:2014-05-14 08:32:48      阅读:245      评论:0      收藏:0      [点我收藏+]

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limbubuko.com,布布扣xabubuko.com,布布扣+bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣f(x)=limbubuko.com,布布扣x+bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣f(x)=Abubuko.com,布布扣

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(xbubuko.com,布布扣0bubuko.com,布布扣)>f(xbubuko.com,布布扣0bubuko.com,布布扣)?εbubuko.com,布布扣0bubuko.com,布布扣>Abubuko.com,布布扣

由$\lim \limits_{x \to

abubuko.com,布布扣+bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣
} f\left( x \right) = A < A + {\varepsilon _0}$及极限的保号性知
?δ>0,?x(a,a+δ),f(x)<A+εbubuko.com,布布扣0bubuko.com,布布扣bubuko.com,布布扣

特别地,取${x_1} \in \left( {a,a + \delta } \right)$,且${x_1} < {x_0}$,则

f(xbubuko.com,布布扣1bubuko.com,布布扣)<A+εbubuko.com,布布扣0bubuko.com,布布扣<f(xbubuko.com,布布扣0bubuko.com,布布扣)bubuko.com,布布扣

由连续函数介值定理知,存在${\xi _1} \in \left( {{x_1},{x_0}} \right)$,使得

f(ξbubuko.com,布布扣1bubuko.com,布布扣)=A+εbubuko.com,布布扣0bubuko.com,布布扣bubuko.com,布布扣

由$\lim \limits_{x \to

+bubuko.com,布布扣bubuko.com,布布扣bubuko.com,布布扣
} f\left( x \right) = A < A + {\varepsilon _0}$及极限的保号性知
?M>a,?x>M,f(x)<A+εbubuko.com,布布扣0bubuko.com,布布扣bubuko.com,布布扣


特别地,取${x_2} \in \left( {M, + \infty } \right)$,且${x_0} < {x_2}$,则

f(xbubuko.com,布布扣2bubuko.com,布布扣)<A+εbubuko.com,布布扣0bubuko.com,布布扣<f(xbubuko.com,布布扣0bubuko.com,布布扣)bubuko.com,布布扣

由连续函数介值定理知,存在${\xi _2} \in \left( {{x_0},{x_2}} \right)$,使得

f(ξbubuko.com,布布扣2bubuko.com,布布扣)=A+εbubuko.com,布布扣0bubuko.com,布布扣bubuko.com,布布扣

由$Rolle$中值定理知,存在$\xi \in \left( {{\xi _1},{\xi _2}} \right)$,使得

fbubuko.com,布布扣bubuko.com,布布扣(ξ)=0bubuko.com,布布扣

05626265,布布扣,bubuko.com

05626265

标签:style   class   code   c   ext   color   

原文地址:http://www.cnblogs.com/ly758241/p/3725570.html

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