Arnol'd's limit and the Lagrange inversion

TL;DR

This paper uses Lagrange inversion to prove Arnol'd's limit involving sine, tangent, arcsine, and arctangent functions, and extends the result to a broader formal power series context.

Contribution

It introduces a novel proof method for Arnol'd's limit using Lagrange inversion and generalizes the result to formal power series.

Findings

The limit equals 1 as x approaches 0.

Lagrange inversion provides a straightforward proof.

The approach extends to a broader class of formal power series.

Abstract

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.