Derivative of Exponential at Zero/Proof 2

Theorem
Let $\exp x$ be the exponential of $x$ for real $x$.

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

Proof
Note that this proof does not presuppose Derivative of Exponential Function.

We use the definition of the exponential as a limit of a sequence:

The right summand converges to zero as $h \to 0$, and so:
 * $\displaystyle \lim_{h \to 0}\frac{\exp h - 1} h = 1$