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Homework Help: Bonus Question Attempt from Calculus Final

  1. Dec 12, 2012 #1
    This was the final bonus question on my first university and I made a serious attempt at it. I know this isn't technically a homework question, so I will understand if it goes unanswered, by founding out if I did this correctly would go a long way to alleviating some of my final exam anxieties.

    1. The problem statement, all variables and given/known data

    f(x)= (1/x)∫0x((1-tan(2t))1/t)dt when x≠0 and k when x=0

    Find the value of k that makes f(x) continuous

    2. Relevant equations

    L'Hopital's Rule, definition of continuity

    3. The attempt at a solution

    limx -> 0 (1/x)∫0x((1-tan(2t))1/t)dt = k

    limx -> 0 (∫0x((1-tan(2t))1/t)dt)/x = k

    By direct substitution, limx -> 0 (∫0x((1-tan(2t))1/t)dt)/x = 0/0

    Since this is an indeterminant form, apply L'Hopital's Rule.

    limx -> 0 ((1-tan(2x))1/x)/1 = k

    limx -> 0 ln( ((1-tan(2x))1/x) ) = ln k

    limx -> 0 (1/x)ln( 1-tan(2x) ) = ln k

    limx -> 0 ln( 1-tan(2x) )/x = ln k

    By direct substition again, limx -> 0 ln( 1-tan(2x) )/x = 0/0.
    Apply L'Hopital's Rule.

    limx -> 0 -(2)sec2(2x)/(1-tan(2x) = ln k

    Direct substition:

    -(2)sec2(0)/(1-tan(0)) = ln k

    -2 = ln k

    k = e-2

    Thanks in advance.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Dec 12, 2012 #2


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    Homework Helper

    That looks right, the answer is right. The short version is
    The integral does not matter (it is the average near zero so just the limit)
    lim (1-tan(2t))^(1/t)=e^ lim -tan(2t)/t
    this follows from lim (1-x)^(1/x)=1/e
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