Homework Help: Calculation of limit

1. Apr 23, 2016

lep11

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. Apr 23, 2016

Math_QED

Yes, since you solved it instantly with it.

3. Apr 23, 2016

lep11

What if we 'don't know' the derivatives of arcsin and arctan?

4. Apr 23, 2016

Math_QED

Then you proof what these derivatives are equal too. It can't be solved in an easier way.

5. Apr 23, 2016

lep11

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?

6. Apr 23, 2016

Ray Vickson

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.