Arnol'd's limit and the Lagrange inversion

We show how to prove by means of the Lagrange inversion the limit of Arnol'd that $$\lim_{x\to0}\frac{\sin(\tan x)-\tan(\sin x)}{\arcsin(\arctan x)-\arctan(\arcsin x)}=1\,.$$ In fact, we obtain a more general result in terms of formal power series.