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 $e$.
Link between $\lim \limits_{n \to \infty} (1+{1/n})^n$ and $\lim \limits_{n \to \infty} (1+{x/n})^n$
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I think you are approaching this in the wrong direction. These identities are not based on the continuous interest model, but are rather definitions for these objects. – Valborg Sep 25 '16 at 21:28
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Is $n$ an integer above? – zhw. Sep 25 '16 at 21:32
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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: $$\begin{align}\lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n &= \lim_{u\rightarrow\infty} \left(1 + \frac{1}{u}\right)^{xu} \\ & = \left[\lim_{u\rightarrow\infty} \left(1 + \frac{1}{u}\right)^{u}\right]^x \\ &= \operatorname{e}^{x}\end{align}$$
Sean Lake
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1I think he was asking about $e^x = \lim \limits_{n \to \infty} (1+{x/n})^n $ not $e^x = \lim \limits_{n \to \infty} (1+{1/n})^{nx} $ – Pentapolis Sep 25 '16 at 21:34
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2OP said he was fine with $\operatorname{e} = \lim_{n\rightarrow\infty} (1 + 1/n)^n$ but was having trouble with $\operatorname{e}^x = \lim_{n\rightarrow\infty} (1 + x/n)^n$. I showed how to get from $\lim_{n\rightarrow\infty} (1 + x/n)^n$ to $\lim_{n\rightarrow\infty} (1 + 1/n)^{xn}$, which, given what he understands, leads right to $\mathrm{e}^x$. – Sean Lake Sep 25 '16 at 21:38
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