March 2010
Euler's formula for ex
We want to prove the identity
(1) |
You might think this argument has made somewhat cavalier use of limits. The following argument, adapted from [1, pg 272], is close to Euler's original. He was even more bold in his use of limits!
"Euler unhesitatingly accepts the existence of both infinitely small and infinitely large numbers, and uses them to such effect that the modern reader's own hesitation must be tinged with envy." [1, pg 272]
Thinking about near , and its gradient, we see that, for infinitely small
Reference
[1] C.H. Edwards, The Historical Development of the Calculus, Springer-Verlag, 1979.