作者Honor1984 (希望愿望成真)
看板trans_math
标题Re: [极限] 请教两题极限的证明
时间Sun Sep 28 17:38:15 2008
※ 引述《phantom997 (囧)》之铭言：
: 1.Prove that lim f(x)=L lim f(x)=M,
: x->c x->c
: then L=M
设 L =/= M
不失一般性下
设 L > M
由 lim f(x) = L 知
选定一任意小之正e < │M-L│/2
可找到一d使│f(x)-L│< e, 0 <│x-c│< d
且同时又满足│f(x)-M│> │L-M│-│f(x)-L│>│M-L│/2
显然与lim f(x) = M极限的定义抵触
x->c
所以矛盾
=> L = M得证
: 2.Suppose that lim f(x)=L and that f(a) exists(though it
: x->a
: maybe different from L).Prove that f is bounded on some
: interval containing a;that is,show that there is an interval
: (c,d) with c<a<d and a constant M such that |f(x)| < M
: =
: for all x in (c,d).
: 谢谢大家罗!!
lim f(x)=L
x->a
选定任意小之e 可找到k使得 │f(x)-L│< e, 0 <│x-a│< k
=> L-e < f(x) < L+e , a-k < x < a+k
令c = a-k
d = a+k
choses M >= max(│L│+ e , │f(a)│)
then │f(x)│<= M for all x in (c,d)
--
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◆ From: 122.124.101.106
1F：推 Qmmm:谢谢~ 202.151.60.129 09/29 11:19
2F：→ Qmmm:推错篇 = = 202.151.60.129 09/29 11:19