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Limit of (arcsen(2x)-2arcsen(x))/x^3 as x->0

  1. Jan 27, 2013 #1
    1. The problem statement, all variables and given/known data
    Limit of (arcsen(2x)-2arcsen(x))/x^3 as x->0


    2. Relevant equations
    arsen(x)'=1/Sqrt(1-x^2)


    3. The attempt at a solution
    It's an 0/0 indeterminancy, to solve it, I had to use L'hopital's rule once simplify de expression, multiplying by the conjugate root. Is there an easier way?
     
  2. jcsd
  3. Jan 27, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    As you don't get rid of the trigonometric functions without l'Hospital, I think this ok.

    $$\frac{\frac{2}{\sqrt{1-(2x)^2}}-\frac{2}{\sqrt{1-(x)^2}}}{x^2} = 2 \frac{1}{\frac{1}{\sqrt{1-(2x)^2}}+\frac{1}{\sqrt{1-(x)^2}}} \frac{\frac{1}{1-(2x)^2}-\frac{1}{1-(x)^2}}{x^2} \to \frac{2}{2} \frac{\frac{1}{1-(2x)^2}-\frac{1}{1-(x)^2}}{x^2} \to \dots$$
     
  4. Jan 27, 2013 #3
    Thanks!
     
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