March 2010

### Euler's formula for *e*^{x}

^{x}

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.