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 $$\underset{x\to 0}{\lim }\frac{\sin (\tan x)-\tan (\sin x)}{\arcsin (\arctan x)-\arctan (\arcsin x)}=1\mathrm{.}$$ In fact, we obtain a more general result in terms of formal power series.