Re: [極限] 請教兩題極限的證明

[Honor1984 (希望願望成真)]

※ 引述《phantom997 (囧)》之銘言：
: 1.Prove that lim  f(x)=L lim f(x)=M,
:              x->c        x->c
: then L=M

設 L =/= M

不失一般性下

設 L > M

由 lim  f(x) = L 知

選定一任意小之正e < │M-L│/2

可找到一d使│f(x)-L│< e, 0 <│x-c│< d

且同時又滿足│f(x)-M│> │L-M│-│f(x)-L│>│M-L│/2

顯然與lim f(x) = M極限的定義牴觸
      x->c

所以矛盾

=> L = M得證

: 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).
: 謝謝大家囉!!

lim f(x)=L
x->a

選定任意小之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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