Derivative of Logarithm at One/Proof 1

Proof
L'Hôpital's Rule gives:
 * $\displaystyle \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \lim_{x \mathop \to c} \frac {\map {f'} x} {\map {g'} x}$

(provided the appropriate conditions are fulfilled).

Here we have:

Thus:
 * $\displaystyle \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = \lim_{x \mathop \to 0} \frac {\paren {1 + x}^{-1} } 1 = \frac 1 {1 + 0} = 1$