# Limit of (Cosine (X) - 1) over X/Proof 2

## Theorem

$\displaystyle \lim_{x \mathop \to 0} \frac {\map \cos x - 1} x = 0$

## Proof

This proof assumes the truth of the Derivative of Cosine Function:

$\cos 0 = 1$
$D_x \left({\cos x}\right) = - \sin x$

and by Derivative of Constant:

$D_x \left({-1}\right) = 0$

So by Sum Rule for Derivatives:

$D_x \left({\cos x - 1}\right) = - \sin x$
$\sin 0 = 0$
$D_x \left({x}\right) = 1$

Thus L'Hôpital's Rule applies and so:

$\displaystyle \lim_{x \mathop \to 0} \frac {\cos x - 1} x = \lim_{x \mathop \to 0} \frac {-\sin x} 1 = \frac {-0} 1 = 0$

$\blacksquare$