Limit of (Cosine (X) - 1) over X at Zero

Theorem

 * $\displaystyle \lim_{x \to 0} \frac {\cos (x) - 1}{x} = 0$

Proof
This proof works directly from the definition of the cosine function:

Alternative Proof 1
This proof assumes the truth of the Derivative of Cosine Function:

We have that:


 * From Basic Properties of Cosine Function: $\cos 0 = 1$;
 * From Derivative of Cosine Function: $D_x \left({\cos x}\right) = - \sin x$ and by Derivative of Constant: $D_x (-1) = 0$. So by Sum Rule for Derivatives $D_x (\cos x - 1) = - \sin x$;
 * By Basic Properties of Sine Function, $\sin 0 = 0$;
 * From Derivative of Identity Function: $D_x \left({x}\right) = 1$.

Thus L'Hôpital's Rule applies and so $\displaystyle \lim_{x \to 0} \frac {\cos x - 1} x = \lim_{x \to 0} \frac {-\sin x} 1 = \frac {-0} 1 = 0$.