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A Kinder, Gentler, Longer L'Hospital Proof

  1. Feb 9, 2008 #1
    I have found the easiest way to understand the proof is to give attention to a lot of particular cases. So in that since, L'Hospital's rule is a compact theorem statement summarizing a handful of situations. That can make it's proof (which part?) hard to remember. Baby Rudin makes it hard to see even the easiest case (below). The "proof" on wiki is just a couple of lines, and isn't really complete. Perhaps this can be posted there... hmm... anyways...

    Last edited: Feb 9, 2008
  2. jcsd
  3. Feb 9, 2008 #2
    I missed some... Any other omissions/mistakes, please point i out or post corrections..

  4. Feb 9, 2008 #3
    One point is that the cases x -> c+ and x -> c- can be merged together and proved as a two sided limit.. In these cases, as I wrote it above I only wrote out the proof for one side.

    As long as I got them all up there, there are 4 distinct approaches (mini#1,#2,#3,#4), and the other cases are all done from the same idea as long as you can make the switch between c < x <= M instead of x >= M, and left/right hand limit etc.

    I get the impression that the proof in baby Rudin is meant to proof LHospital via only two cases, except the idea involved is more difficult, whereas the proofs I gave above are all very straightforward. I felt that using a "difficult idea" to simplify it to two cases (instead of just one) is not worth the price in comparison to 4 easy ideas for 4 cases.
    Last edited: Feb 9, 2008
  5. Feb 10, 2008 #4
    I should finish off by organizing around the ideas, a 4 part L'Hospital.

    OK That's enough!!!
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