Crafty solutions to the following limit The following problem came up at dinner, I know some ways to solve it but they are quite ugly and as some wise man said: There is no place in the world for ugly mathematics.
These methods are using l'Hôpital, but that becomes quite hideous very quickly or by using series expansions.
So I'm looking for slick solutions to the following problem:

Compute $\displaystyle \lim_{x \to 0} \frac{\sin(\tan x) - \tan(\sin x)}{\arcsin(\arctan x) - \arctan(\arcsin x)}$.

I'm curious what you guys will make of this.

By Taylor's series we obtain that

*

*$\sin (\tan x)= \sin\left(x + \frac13 x^3 + \frac2{15}x^5+ \frac{17}{315}x^7+O(x^9) \right)=x + \frac16 x^3 -\frac1{40}x^5 - \frac{55}{1008}x^7+O(x^9)$
and similarly

*

*$\tan (\sin x)= x + \frac16 x^3 -\frac1{40}x^5 - \frac{107}{5040}x^7+O(x^9)$

*$\arcsin (\arctan x)= x - \frac16 x^3 +\frac{13}{120}x^5 - \frac{341}{5040}x^7+O(x^9)$

*$\arctan (\arcsin x)= x - \frac16 x^3 +\frac{13}{120}x^5 - \frac{173}{5040}x^7+O(x^9)$
which leads to the result.
