Derivative of Exponential at Zero/Proof 3

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

$\blacksquare$