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Calculate limit using taylor series

  1. Sep 10, 2012 #1
    1. The problem statement, all variables and given/known data

    Calculate: $$ \displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos \left( 1-\cos x \right)}{{{x}^{4}}}$$

    2. Relevant equations



    3. The attempt at a solution

    Using Taylor series I have:

    $$ \displaystyle f'\left( x \right)=\sin \left( 1-\cos x \right)\sin x$$

    $$\displaystyle f'\left( 0 \right)=0$$

    but as you can see, now it turns very tedious to continue differentiating. Do I have to keep differentiating until I get something? Or is there any quicker way to compute this?

    Thanks
     
  2. jcsd
  3. Sep 11, 2012 #2
    There is. All you need to use is the Taylor series for cos(x). Also remember that since you're taking limit of x->0, you only need to carry with you the terms of the lowest non-trivial order.
     
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