# Limit of (Cosine (X) - 1) over X/Proof 4

$\displaystyle \lim_{x \mathop \to 0} \frac {\map \cos x - 1} x = 0$
 $\displaystyle \frac {\cos x - 1} x$ $=$ $\displaystyle \frac {\cos x - \cos 0} x$ Cosine of Zero is One $\displaystyle$ $\to$ $\displaystyle \left.{\dfrac {\mathrm d} {\mathrm dx} \cos x}\right \vert_{x \mathop = 0}$ as $x \to 0$, from definition of derivative at a point $\displaystyle$ $=$ $\displaystyle \sin x \vert_{x \mathop = 0}$ Derivative of Cosine Function $\displaystyle$ $=$ $\displaystyle 0$ Sine of Zero is Zero
$\blacksquare$