Derivative of Exponential at Zero/Proof 3

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Theorem

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


Then:

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


Proof

\(\ds \frac {e^x - 1} x\) \(=\) \(\ds \frac {e^x - e^0} x\) Exponential of Zero
\(\ds \) \(\to\) \(\ds \valueat {\dfrac \d {\d x} e^x} {x \mathop = 0} {}\) Definition of Derivative of Real Function at Point, as $x \to 0$
\(\ds \) \(=\) \(\ds \valueat {e^x} {x \mathop = 0} {}\) Derivative of Exponential Function
\(\ds \) \(=\) \(\ds 1\) Exponential of Zero

$\blacksquare$