Euler's Limit -- Revisited

TL;DR

This paper revisits Euler's limit, demonstrating that for sequences with specific asymptotic relations, the limit of a particular exponential expression converges to e raised to a constant.

Contribution

It provides a concise proof confirming the limit behavior for sequences with asymptotic proportionality, extending Euler's limit to broader sequence classes.

Findings

The limit equals e^{k} under given conditions.

Asymptotic proportionality determines the exponential limit.

The result generalizes classical Euler's limit to sequences with specific growth.

Abstract

The aim of this short note is that if $\{ a_{n}\}$ and $\{ b_{n}\}$ are two sequences of positive real numbers such that $a_{n}\to +\infty$ and $b_n$ satisfying the asymptotic formula $b_n\sim k\cdot a_{n}$, where $k>0$, then $\lim\limits_{n\to\infty}\left(1+\frac{1}{a_{n}}\right)^{b_{n}}= e^{k}$.