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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

    lurflurf

    User Avatar
    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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