Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$ At this link someone asked how to prove rigorously that
$$
\lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x.
$$
What good intuitive arguments exist for this statement?  
Later edit: . . . where $e$ is defined as the base of an exponential function equal to its own derivative.
I will post my own answer, but that shouldn't deter anyone else from posting one as well.

Another way of looking at it:
Let
$$f_n(x) = \left(1+\frac{x}{n}\right)^n$$ and we are interested in $f(x) = \lim_{n \to \infty} f_n(x)$
Now,
$$f_n(x) f_n(y) = \left(1+\frac{x}{n}\right)^n\left(1+\frac{y}{n}\right)^n$$
$$ = \left(1+\frac{x+y +\frac{xy}{n}}{n}\right)^n = f_n\left(x+y +\frac{xy}{n}\right)$$
Thus as $n \to \infty$, we probably have that
$$f(x)f(y) = f(x+y)$$
and so we can expect $f(x)$ to be exponential.
