Euler's Limit -- Revisited

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