TITLE: Evaluating $\lim_{x\to0}\frac{1-\cos(x)}{x}$
QUESTION [11 upvotes]: $$\lim_{x\to0}\frac{1-\cos(x)}{x}$$
Could someone help me with this trigonometric limit? I'm trying to evaluate it without L'Hôpital's rule and derivation.

REPLY [3 votes]: Using the definition of derivative: $$\lim\limits_{x \to 0} \frac{1-\cos(x)}{x}= - \lim\limits_{x \to 0} \frac{\cos(x)-\cos(0)}{x-0}=- \cos'(0)=-\sin(0)=0$$