Nilai \( \displaystyle \lim_{x \to 0} \ \frac{\sin x + \sin 5x}{6x} = \cdots \)
- 2
- 1
- 1/2
- 1/3
- -1
(UN SMA IPA 2010)
Pembahasan:
\begin{aligned} \lim_{x \to 0} \ \frac{\sin x + \sin 5x}{6x} &= \lim_{x \to 0} \ \frac{2 \sin \left( \frac{x+5x}{2} \right) \cos \left( \frac{x-5x}{2} \right)}{6x} \\[8pt] &= \lim_{x \to 0} \ \frac{2 \sin 3x \cos (-2x)}{6x} \\[8pt] &= 2 \cdot \lim_{x \to 0} \ \frac{\sin 3x}{6x} \cdot \lim_{x \to 0} \ \cos(-2x) \\[8pt] &= 2 \cdot \frac{3}{6} \cdot \cos(-2 \cdot 0) = \frac{6}{6} \cdot \cos 0 \\[8pt] &= 1 \cdot 1 = 1 \end{aligned}
Jawaban B.