Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof of the L'Hôpital Rule for the Indeterminate Form $\frac{\infty}{\infty}$

  1. May 20, 2012 #1
    I ask for the Proof of the L'Hôpital Rule for the Indeterminate Form [tex]\frac{\infty}{\infty}[/tex] utilising the Rule for the form [tex]\frac{0}{0}[/tex]

    The Theorem: Let [tex]f,g:(a,b)\to \mathbb{R}[/tex] be two differentiable functions such as that:
    [tex]\forall x\in(a,b)\ \ g(x)\neq 0\text{ and }g^{\prime}(x)\neq 0[/tex] and [tex]\lim_{x\to a^+}f(x)=\lim_{x\to a^+}g(x)=+\infty[/tex]
    If the limit $$\lim_{x\to a^+}\frac{f^{\prime}(x)}{g^{\prime}(x)}$$ exists and is finite, then
    $$\lim_{x\to a^+}\frac{f(x)}{g(x)}=\lim_{x\to a^+}\frac{f^{\prime}(x)}{g^{\prime}(x)}$$

    My attempt:
    Since [tex]\lim_{x\to a^+}f(x)=+\infty[/tex], $$\exists \delta>0:a<x<a+\delta<b\Rightarrow f(x)>0\Rightarrow f(x)\neq 0$$
    Let [tex]F,G:(a,a+\delta)[/tex] [tex]F(x)=\frac{1}{f(x)}[/tex], [tex]G(x)=\frac{1}{g(x)}[/tex] Then by the hypothesis [tex]\lim_{x\to a^+}F(x)=\lim_{x\to a^+}G(x)=0[/tex] $$\forall x\in(a,b)\ \ G(x)\neq 0\text{ and }G^{\prime}(x)=-\frac{1}{g^2(x)}g^{\prime}(x)\neq 0$$
    The question is, does the limit $$\lim_{x\to a^+}\frac{F^{\prime}(x)}{G^{\prime}(x)}=\lim_{x\to a^+}\frac{-\frac{1}{f^2(x)}f^{\prime}(x)}{-\frac{1}{g^2(x)}g^{\prime}(X)}=\lim_{x\to a^+}\frac{g^2(x)f^{\prime}(x)}{f^2(x)g^{\prime}(x)}$$ exist?

    The limit $$\lim_{x\to a^+}\frac{f^{\prime}(x)}{g^{\prime}(x)}$$ exists by the hypothesis but we don't know if the limit $$\lim_{x\to a^+}\frac{g^2(x)}{f^2(x)}$$ exists to deduce that the limit $$\lim_{x\to a^+}\frac{F^{\prime}(x)}{G^{\prime}(x)}$$ exists and use the L'Hôpital Rule for the form [tex]\frac{0}{0}[/tex]
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted

Similar Discussions: Proof of the L'Hôpital Rule for the Indeterminate Form $\frac{\infty}{\infty}$
  1. Bernoulli's Rule (Replies: 2)

  2. Proof Writing (Replies: 3)