\lim_{x\to 0}\frac{\sin x}{x}$ and the definition of $\pi$

TL;DR

This paper presents two new proofs for the limit of sin x over x as x approaches zero, exploring its connection to the definition of pi, building on historical methods involving trigonometric addition formulas.

Contribution

It introduces novel proofs for the limit of sin x over x and discusses its relationship with the mathematical constant pi, expanding on classical derivations.

Findings

Two new proofs for the limit of sin x over x as x approaches zero.

Clarification of the connection between this limit and the definition of pi.

Historical context linking trigonometric addition formulas to the limit.

Abstract

Leopold Vietoris and Guido Hoheisel showed how the existence of $\lim_{x\to 0}\frac{\sin x}{x}$ can be derived from the trigonometric addition formulas. In this article two new proofs for this result are given. In addition it is discussed how this limit is related to the definition of $\pi$.