Re: 微積分證明題
※ 引述《coffee1205 (佐岸)》之銘言：
: ※ 引述《coffee1205 (佐岸)》之銘言：
: : Let f(x)=x/|x|.Prove that limitf(x) does not exist?
: : x->0
: : Hint:Show that no number L qualifies as the limit because
: : there always some x such that |x| < delta,but |f(x)-L|大於等於1/2
: : ,no matter how small delta is taken.
以下錯誤.
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首先我們斷言,lim f(x)極限不存在. 若不然,我們設其值為L(LεR),且取ε=1/2
x→0
df
即lim f(x)=L <==> for ε=1/2 , there exists a δ>0 such that
x→0
0<|x-0|<δ => |f(x)-L|<ε=1/2 ---(*)
則存在δ>0,故對此δ而言,顯然如下
0<|x-0|<δ ==> |f(x)-L| = | x/|x|-L | <= (|x|/|x|) +|-L| = 1+|L| --(**)
故由(**)知,|f(x)-L| <= 1+|L| < 1/2 顯然矛盾
故知假設錯誤,所以lim f(x)極限不存在.
x→0
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※ 編輯: zptdaniel 來自: 123.194.97.168 (10/19 18:57)
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