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Finding Limit as x-> 0 of sin and cos equation

  1. Feb 13, 2014 #1

    Bro

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    1. The problem statement, all variables and given/known data
    lim (-x + sin(sinx))/(x(-1 + cos(sinx)))
    x-> 0

    2. Relevant equations
    sinθ/θ=1 as θ-> 0
    Squeeze Theorem?


    3. The attempt at a solution
    Well I've tried doing L'Hopital's rule, but to no avail. Each subsequent derivative just makes the equation nastier and more convoluted.
    Eg. lim d/dx(-x + sin(sinx))/d/dx(x(-1 + cos(sinx)))=-1 + cosx(cos(sinx))/(-1 + cos(sinx) - xcosx(sin(sinx)))=the next one is even longer and nothing jumps out at me for a possible solution.
    I've also attempted to rewrite the equation but again to no avail.
    I've also thought about squeeze theorem, but don't know how to apply it here.
    Any ideas?
     
  2. jcsd
  3. Feb 13, 2014 #2

    Dick

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    Keep going till you stop getting 0/0. It will crack at the third derivative.
     
  4. Feb 13, 2014 #3

    haruspex

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    As long as you're careful you can simplify some before proceeding.
    cos(sin(x)) is going to be very much like cos(x). The discrepancy is of order x4.
    In the denominator, the leading term of -1 + cos(sinx) is O(x2). You can throw away all higher order terms in the denominator, and as long as the result is not zero the simplification is justified.
     
  5. Feb 13, 2014 #4

    Bro

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    Ok, but what is O?
     
  6. Feb 13, 2014 #5

    haruspex

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    It's "big O" notation. http://en.wikipedia.org/wiki/Big_O_notation. E.g. in the Taylor expansion sin(x) = x - x3/3! + ... we can write sin(x) = x + O(x3). That is, sin(x) is like x plus terms of order x3 and beyond (for small x).
    In the present problem x will go to zero. If the numerator can be reduced to Axn+O(xn+1) then as x approaches the limit we can simplify it to Axn. The xn+1 and beyond terms will become irrelevant. Likewise the denominator.
     
  7. Feb 13, 2014 #6

    Bro

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    Alright perfect. Thank you both for your help!
     
  8. Feb 14, 2014 #7

    Curious3141

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    Taylor series applied judiciously gives a quick solution.
     
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