Link between $\lim \limits_{n \to \infty} (1+{1/n})^n$ and $\lim \limits_{n \to \infty} (1+{x/n})^n$

Question

I understand that the intuition behind $e = \lim \limits_{n \to \infty} (1+{1/n})^n$, which can be understood as a continuous interest of 100% over time. However I'm having troubles understanding $e^x = \lim \limits_{n \to \infty} (1+{x/n})^n$ with the same intuition, specifically at how an 100x% interest turns into an exponentiation of $e$.

Answer 1

I don't have an intutive way to understand it, but there is a simple algebraic way. Do the substitution $n = xu$ and the limit becomes: $$\begin{align}\lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n &= \lim_{u\rightarrow\infty} \left(1 + \frac{1}{u}\right)^{xu} \\ & = \left[\lim_{u\rightarrow\infty} \left(1 + \frac{1}{u}\right)^{u}\right]^x \\ &= \operatorname{e}^{x}\end{align}$$