L'hospital's rule

How to prove it? I've watched a tv program before and I vaguely remember the proof consists of 4 parts, which includes the mean value theorem, Cauchy's mean value theorem and the other 2 theorems. I haven't learnt mean value theorem at that time and I didn't understand it, perhaps I can now.
 
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Check out http://mathforum.org/library/drmath/view/53340.html. It's a bit too large to type in the forum. The proof that uses Cauchy's mean value theorem begins

Code:
Date: 12/23/98 at 11:43:47
From: Doctor Rob
Subject: Re: Proof of L'hopital's rule
The first proof uses Taylor expansions. The second uses Cauchy's MVT.
 

Dx

Originally posted by KL Kam
How to prove it? I've watched a tv program before and I vaguely remember the proof consists of 4 parts, which includes the mean value theorem, Cauchy's mean value theorem and the other 2 theorems. I haven't learnt mean value theorem at that time and I didn't understand it, perhaps I can now.
Visual Calculus
 

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