Re: [極限] x*lnx , x→0+

[JohnMash (Paul)]

※ 引述《znmkhxrw (QQ)》之銘言：
:  lim  x*lnx = ?
: x→0+

      Let lnx = -u

      then x=e^{-u}

      x→0+ becomes u→∞

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◆ From: 112.104.129.200
推 znmkhxrw    :請問一下lim x→+inf x/e^x = 0 的詳細證明是??
→ znmkhxrw    :用L'hospital
→ znmkhxrw    :我的意思是 lim x→+inf f(x)/g(x) 的L'怎麼證??

  Forget it if you are not interested in mathematical analysis.


--------------------------------------------------
   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)]
--------------------------------------------------

   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

※ 編輯: JohnMash        來自: 112.104.142.5        (07/06 10:34)