\begin{aligned}
\lim_{n\to\infty}\Big(1+\frac{k}{n}\Big)^n &= \lim_{n\to\infty}\bigg(\Big(1+\frac{1}{n/k}\Big)^{n/k}\bigg)^k \\
&= \lim_{m\to\infty}\bigg(\Big(1+\frac{1}{m}\Big)^m\bigg)^k \\
&= \bigg(\lim_{m\to\infty}\Big(1+\frac{1}{m}\Big)^m\bigg)^k \\
&= e^k
\end{aligned} |