[Fly]

Mathematicians were looking for a function that is equal to its own derivative. They narrowed down the search to functions of the form f(x) = a^x, where a is real.

For a fixed x,

f'(x) = lim (1/h)( f(x+h) - f(x) ) where h--> infinity

a^x = lim (1/h)( a^(x+h) - a^x ) where h---> infinity

factoring out a^x from the right hand side

a^x = a^x * lim (1/h)( a^x - 1) where h--> infinity

1 = lim (1/h) (a^x - 1 ) where h--->infinity

e is defined to be the unique value of a such that the equation above is true. You can massage the equation above and substitute h = 1/n to get the definition provided by previous posters.