Derivative of Exponential at Zero/Proof 1

Proof
For all $x \in \R$:


 * $\exp 0 - 1 = 0$ from Exponential of Zero


 * $\map {D_x} {\exp x - 1} = \exp x$ from Sum Rule for Derivatives


 * $D_x x = 1$ from Derivative of Identity Function.

Its prerequisites having been verified, Corollary 1 to L'Hôpital's Rule yields immediately:


 * $\displaystyle \lim_{x \mathop \to 0} \frac {\exp x - 1} x = \lim_{x \mathop \to 0} \frac {\exp x} 1 = \exp 0 = 1$