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Homework Help: A limit problem

  1. Dec 21, 2006 #1
    1. The problem statement, all variables and given/known data
    f(x)=(tan(x)/x)^(1/(x^2)) it asks the limit of this function when x goes to 0


    2. Relevant equations



    3. The attempt at a solution

    i have tried to take the ln of the two sides than used the l'hopital rule but with that way i could not reach anything. pls help me
     
  2. jcsd
  3. Dec 21, 2006 #2

    StatusX

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    What exactly went wrong when you tried that?
     
  4. Dec 21, 2006 #3
    Develop the tan(x) in a series first. Using the actual tan(x) always gives me infinity divided by zero.

    The Taylor series of tan(x) around zero is valid for |x| < pi/2 so...


    marlon
     
    Last edited: Dec 21, 2006
  5. Dec 21, 2006 #4

    dextercioby

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    If the limit is NOT 0, then Marlon's suggested method leads to an erroneous result.

    Daniel.
     
  6. Dec 21, 2006 #5
    Actually, YES, you are right. Actually, i don't know how to solve it so i am gonna say it's indefinite :rofl:

    marlon
     
  7. Dec 21, 2006 #6

    dextercioby

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    I get infinity, plus or minus, depending on whether the limit is approaching 0 from below or from above.

    Daniel.
     
  8. Dec 21, 2006 #7
    Yeah, (1+x^2)^(1/x^2) for x--> 0 (after doing the Taylor thing) gives me this : 1 + x > 1 and the power gets bigger if x gets towards 0, so you are EVOLVING towards infinity but what i cannot achieve is prove that the value is actually infinite

    Also, if x is coming from te negative side, you are again evolving toward positive infinity because tanx/x = 1 + (x^2)/3 + ... with all positive powers !!!

    marlon
     
    Last edited: Dec 21, 2006
  9. Dec 21, 2006 #8

    dextercioby

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    I ****ed up the derivatives, i've had too much to drink at the party, apparently. By using the method suggested by Marlon, i now get e^{3}.

    I won't go through that l'Hopital again. :d

    Daniel.
     
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