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

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Theorem

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


Proof

\(\displaystyle \frac {\cos x - 1} x\) \(=\) \(\displaystyle \frac {\cos x - \cos 0} x\) 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

$\blacksquare$