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Homework Help: L'Hospital of e^(1/x) as limit x-> 0- and 0+

  1. Aug 8, 2011 #1
    e^(1/x) as limit x-> 0- and 0+

    1. The problem statement, all variables and given/known data
    lt e^(1/x) = 0


    lt e^(1/x) = +( infinity)

    2. Relevant equations

    3. The attempt at a solution

    let y = e^(1/x)
    ln y = (1/x) ln e

    lt (ln e)/x

    applying L'Hospitals rule = 0

    How to procedd with the lt x-> 0+

    Kindly help me.
    Last edited: Aug 8, 2011
  2. jcsd
  3. Aug 8, 2011 #2


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    You're not using l'hopital's rule correctly. To use l'hopital, you need +-inf/inf, or 0/0, neither of which you have. It doesn't look like you're even taking the derivatives? You're using a different technique.

    In any case, why are you trying to use l'hopital's rule for this? You can evaluate it directly. For:

    [tex]\lim_{x \to 0^{-}}e^{\frac{1}{x}}[/tex]

    What is the exponent approaching? What is e raised to this? Do the same thing for the other problem.
  4. Aug 8, 2011 #3
    The question was just to prove the limit. So thats why i used L'hospitals rule.
  5. Aug 8, 2011 #4


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    Using l'hopital's rule is no more legit than evaluating the limit directly. If you want to prove the limit, you need to use the epsilon-delta definition of limits.
  6. Aug 8, 2011 #5
    Thanks for the explanation
  7. Aug 8, 2011 #6


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    I wouldn't say you have to use the delta-epsilon definition. Evaluating the limit is fine, but you just need to justify each step.

    While you can use L'Hopital's rule as part of a proof, you can only do so when it's applicable to the particular problem, which it isn't in this case.
  8. Aug 9, 2011 #7
    Also, it is L'Hopital's rule not L'Hospital's rule. That way people won't look at you funny when you asked for help on the problem :wink:
  9. Aug 9, 2011 #8


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    Wolfram Mathworld http://mathworld.wolfram.com/LHospitalsRule.html: [Broken]
    "Note that l'Hospital's name is commonly seen spelled both "l'Hospital" (e.g., Maurer 1981, p. 426; Arfken 1985, p. 310) and "l'Hôpital" (e.g., Maurer 1981, p. 426; Gray 1997, p. 529), the two being equivalent in French spelling."
    Last edited by a moderator: May 5, 2017
  10. Aug 9, 2011 #9


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    It depends how technical you want to be. I can eyeball something like:

    [tex]\lim_{x \to 0} x^2[/tex]

    and tell you that the limit is 0, or I can simply plug in values that get closer and closer to 0. Did I prove it? No. However, just about anyone can see that the limit is 0. If you want to actually prove it mathematically, the only way I can see this possible is by using e-d. If I'm overloking something though, I'm all ears.


    I think I may know what you're talking about. For a lot of problems, you can take exploit continuity. In my example (and a lot of examples), you can. This of course assumes that you're allowed to assume the function is continuous at the point in question (which requires e-d and evaluating the limit there anyways). However, whenever I see "prove the limit" in a calculus book, I never think of evaluating it.
    Last edited: Aug 9, 2011
  11. Aug 9, 2011 #10
    I agree fully, but the rules for french spelling have changed.

    "In the 17th and 18th centuries, the name was commonly spelled "l'Hospital", however, French spellings have been altered: the silent 's' has been dropped and replaced with the circumflex over the preceding vowel." - http://en.wikipedia.org/wiki/Guillaume_de_l'Hôpital

    Similar to forêt and forest... I actually wonder if the name was changed posthumously or if he started life as l'Hôpital, any ideas?

    Either way, the pronounciation should definitely not include the 's'. That's why I mentioned it, so that the OP doesn't go around asking colleagues or professors about the l'Hospital rule using the english pronounciation for hospital, which may cause confusion or get a giggle out of said colleague or professor :wink:
    Last edited by a moderator: May 5, 2017
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