My Cousin's Challenge: Solve a Tough Limit Problem

[davedave]

My cousin challenges me with a tough limit problem.

Compute the limit as x->0 of (sin(tanx)-tan(sinx))/(arcsin(arctanx)-arctan(arcsinx)).

This limit seems impossible to do.

I used L'Hopital's rule but it didn't help at all.

Can anyone find a way to do it?

 

[hamster143]

Expand the numerator and the denominator as Taylor series up to the first significant term.