Limit of (Cosine (X) - 1) over X at Zero/Proof 4
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Theorem
- $\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$
Proof
| \(\ds \frac {\cos x - 1} x\) | \(=\) | \(\ds \frac {\cos x - \cos 0} x\) | Cosine of Zero is One | |||||||||||
| \(\ds \) | \(\to\) | \(\ds \valueat {\dfrac \d {\d x} \cos x} {x \mathop = 0}\) | as $x \to 0$, from Definition of Derivative of Real Function at Point | |||||||||||
| \(\ds \) | \(=\) | \(\ds \bigvalueat {\sin x} {x \mathop = 0}\) | Derivative of Cosine Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) | Sine of Zero is Zero |
$\blacksquare$