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

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

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

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

Alternative Proof 2
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