A Generalization of Euler’s Limit

Summary

The famous Euler’s limit is $\underset{n\to \infty }{\lim }{(\frac{n+1}{n})}^n=e.$ In this note, we observe yet another generalization of Euler’s limit as follows:

Let $\left\{a_n\right\}$ and $\left\{b_n\right\}$ be two sequence of real numbers such that a_n > 1 and $a_n\to 1$ and b_n is satisfying the asymptotic formula $b_n\sim \frac{k}{a_n-1}$, where k > 0, then $\underset{n\to \infty }{\lim }{a}_n^{b_n}=e^k.$

2020 Mathematics Subject Classification::

• 40A05
• 11B83