Limit of (Cosine (X) - 1) over X at Zero/Proof 3
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Theorem
- $\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$
Proof
| \(\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x\) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac {\paren {\cos x - 1} \paren {\cos x + 1} } {x \paren {\cos x + 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac {\cos^2 x - 1} {x \paren {\cos x + 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac {-\sin^2 x} {x \paren {\cos x + 1} }\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\lim_{x \mathop \to 0} \frac {\sin x} x} \paren {\lim_{x \mathop \to 0} \frac {-\sin x} {\cos x + 1} }\) | Product Rule for Limits of Real Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \times \paren {\lim_{x \mathop \to 0} \frac{-\sin x} {\cos x + 1} }\) | Limit of $\dfrac {\sin x} x$ at Zero | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\ds \lim_{x \mathop \to 0} \paren {-\sin x} } {\ds \lim_{x \mathop \to 0} \paren {\cos x + 1} }\) | Quotient Rule for Limits of Real Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 0 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) |
$\blacksquare$