作者JohnMash (Paul)
看板Math
标题Re: [极限] x*lnx , x→0+
时间Wed Jul 6 08:31:18 2011
※ 引述《znmkhxrw (QQ)》之铭言：
: lim x*lnx = ?
: x→0+
Let lnx = -u
then x=e^{-u}
x→0+ becomes u→∞
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※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 112.104.129.200
1F：推 znmkhxrw :请问一下lim x→+inf x/e^x = 0 的详细证明是?? 07/06 09:05
2F：→ znmkhxrw :用L'hospital 07/06 09:05
3F：→ znmkhxrw :我的意思是 lim x→+inf f(x)/g(x) 的L'怎麽证?? 07/06 09:06
Forget it if you are not interested in mathematical analysis.
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if f(a)=g(a)=0
f(x)/g(x)=[f(x)-f(a)]/[g(x)-g(a)]
=[(f(x)-f(a))/(x-a)] / [(g(x)-g(a))/(x-a)]
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if f(∞)=∞,g(∞)=∞
Let u=1/x, h(x)=1/f(x), p=1/g(x)
h'(x)=-1/f^2(x) * f'(x)
p'(x)=-1/g^2(x) * g'(x)
then
f(x)/g(x)=p(x)/h(x)=p(1/u)/h(1/u)
lim_{x→∞} f(x)/g(x)
= lim_{u→0+} p(1/u)/h(1/u)
= lim_{u→0+} [p'(1/u) * (-1/u^2)] / [h'(1/u) * (-1/u^2)]
= lim_{u→0+} p'(1/u) / h'(1/u)
=lim_{x→∞} p'(x)/h'(x)
=lim_{x→∞} g'(x)/f'(x) * f^2(x)/g^2(x)
= lim_{x→∞} g'(x)/f'(x) * [lim_{x→∞} f(x)/g(x)]^2
hence
lim_{x→∞} f(x)/g(x)=lim_{x→∞} f'(x)/g'(x)
or
lim_{x→∞} f(x)/g(x)=0
※ 编辑: JohnMash 来自: 112.104.142.5 (07/06 10:34)