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Calculation of limit

  1. Apr 23, 2016 #1
    1. The problem statement, all variables and given/known data
    Calculate ##\lim_{x \rightarrow 0} \frac{arctanx}{arcsinx}## 'rigoriously'.
    3. The attempt at a solution
    What's the best approach? L'Hospitals rule?

    ##\lim_{x \rightarrow 0} \frac{arctanx}{arcsinx}=\lim_{x \rightarrow 0} \frac{\sqrt{1-x^2}}{x^2+1} =1##
     
  2. jcsd
  3. Apr 23, 2016 #2

    Math_QED

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    Yes, since you solved it instantly with it.
     
  4. Apr 23, 2016 #3
    What if we 'don't know' the derivatives of arcsin and arctan?
     
  5. Apr 23, 2016 #4

    Math_QED

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    Then you proof what these derivatives are equal too. It can't be solved in an easier way.
     
  6. Apr 23, 2016 #5
    I should check the conditions for l' Hospitals rule first.
    I am thinking whether I can assume we know the derivatives or begin with calculating the derivatives first?
    It's kinda re-inventing the wheel though?

    How about applying taylor series of arcsin and arctan?
     
  7. Apr 23, 2016 #6

    Ray Vickson

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    How would you find the Taylor series without knowing the derivatives?

    Nobody is re-inventing the wheel here. If you know the derivatives (or can find them easily) then l'Hospital's rule is useful; otherwise, it does you no good. In your case you know the derivatives, so l'Hospital works like a charm.
     
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