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So I am currently going over some old calculus concepts and ran cross the exponential function expressed as a limit:

$$e^x = \lim_{n\rightarrow \infty}\left(\!1+\frac{x}{n}\!\right)^n $$

Where exactly does this come from? How can we show that this is true?

Additionally, I have also seen this:

$$e^x = \lim_{n\rightarrow 0}(1+xn)^{\frac{1}{n}} $$

How can we show that this is true?

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    $n$ as a natural number cannot converge to $0$. About the second limit. – 2017-01-04
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    @SachpazisStelios it's just a dummy variable, it doesn't need to be an integer – 2017-01-04
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    How do you define $e^x$? – 2017-01-04
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    Nice question, but already addressed [here](http://math.stackexchange.com/q/882741/152225). – 2017-01-04

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