# Calculation of limit

• lep11

## Homework Statement

Calculate ##\lim_{x \rightarrow 0} \frac{arctanx}{arcsinx}## 'rigoriously'.

## 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##

Yes, since you solved it instantly with it.

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

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

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

Yes, since you solved it instantly with it.
I should check the conditions for l' Hospitals rule first.
Then you proof what these derivatives are equal too. It can't be solved in an easier way.
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?

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?

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.