Nilai \( \displaystyle \lim_{x \to 0} \ \frac{2x \sin 3x}{1-\cos 6x} = \cdots \)
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(UN SMA IPA 2007)
Pembahasan:
\begin{aligned} \lim_{x \to 0} \ \frac{2x \sin 3x}{1-\cos 6x} &= \lim_{x \to 0} \ \frac{2x \sin 3x}{1-\cos 2(x)} \\[8pt] &= \lim_{x \to 0} \ \frac{2x \sin 3x}{1-(\cos^2 3x - \sin^2 3x)} \\[8pt] &= \lim_{x \to 0} \ \frac{2x \sin 3x}{1-(1 - \sin^2 3x - \sin^2 3x)} \\[8pt] &= \lim_{x \to 0} \ \frac{2x \sin 3x}{2 \sin^2 3x} = \lim_{x \to 0} \ \frac{x}{\sin 3x} \\[8pt] &= \frac{1}{3} \end{aligned}
Jawaban D.