Question

How is the number e defined in calculus?

Answer 1

MANY ways. Let me list them:

\[\lim_{n \to \infty}\left(1+\frac{1}{n}\right)^{n}\]
\[\lim_{x \to 0}(1+x)^{\frac{1}{x}}\]
\[\sum_{k \geq 0}\frac{1}{k!}\]

\[\int_{1}^{e}\frac{1}{t}dt=1\]
\[\frac{d}{dt}log_{e}t=\frac{1}{t}\]
\[\frac{d}{dt}e^t=e^t\]

There are many, many more.

See more here: http://en.wikipedia.org/wiki/Representations_of_e