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Why is this limit not zero?

  1. Jun 19, 2013 #1
    1. The problem statement, all variables and given/known data

    Why is the limit of [tex]ln(x)/arctan(x)[/tex] as x approaches 0 from the right, not zero?

    2. Relevant equations



    3. The attempt at a solution

    I used L'hopital's rule and got zero. But question specifically states that is not the answer.
     
  2. jcsd
  3. Jun 19, 2013 #2

    jbunniii

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    $$\lim_{x \rightarrow 0^+} \ln(x) = -\infty$$
    whereas
    $$\lim_{x \rightarrow 0^+} \arctan(x) = 0$$
    so L'Hopital's rule does not apply.
     
  4. Jun 19, 2013 #3

    HallsofIvy

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    As jbunniii said, L'Hopital's rule does not apply here. Notice that If x= .000001, we have
    [tex]\frac{ln(.000001)}{arctan(.000001)}= \frac{-13.8155}{.000001}= -13815510[/tex]
    not anywhere near 0!
     
  5. Jun 19, 2013 #4
    Thank you!

    But, that's all there is to it? I don't have to simplify anything? Just plug in values?
     
  6. Jun 19, 2013 #5

    HallsofIvy

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    No, I didn't say that. In order to prove that a limit is a specific number you have to prove that it gets arbitrarily close to that number. My point was that for x very close to 0, the function value is very far away from 0. It is theoretically possible that a function would turn from being around -13 million at x= .000001 to 0 at x= 0, but that would be a very strange function!
     
  7. Jun 19, 2013 #6

    haruspex

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    It's so obviously unbounded that it's rather a strange question. Are you sure you've stated it correctly?
     
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