05626

设[mathop {lim }limits_{x o egin{array}{*{20}{c}}
{{a^ + }}
end{array}} fleft( x ight) = mathop {lim }limits_{x o egin{array}{*{20}{c}}
{ + infty }
end{array}} fleft( x ight) = A]

其中$A$是有限数或$pm infty $

若$fleft( x ight) = A$,则结论显然成立；若$fleft( x ight) e A$,则存在${x_0} in left( {a, + infty } ight)$,使得$fleft( {{x_0}} ight) e A$.

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

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

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

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

由$lim limits_{x o egin{array}{*{20}{c}}
{ + infty }
end{array}} fleft( x ight) = A < A + {varepsilon _0}$及极限的保号性知
[exists M > a,forall x > M,有fleft( x ight) < A + {varepsilon _0}]

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

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

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