Nilai \( \displaystyle \lim_{x \to 0} \ \frac{1-\cos^2 x}{x^2 \cot (x+\frac{\pi}{3})} = \cdots \)
- -1
- 0
- 1
- \( \frac{\sqrt{2}}{2} \)
- \( \sqrt{3} \)
(SNMPTN 2012)
Pembahasan:
\begin{aligned} \lim_{x \to 0} \ \frac{1-\cos^2 x}{x^2 \cot (x+\frac{\pi}{3})} &= \lim_{x \to 0} \ \frac{\sin^2 x}{x^2 \cot (x+\frac{\pi}{3})} \\[8pt] &= \lim_{x \to 0} \ \frac{\sin^2 x}{x^2} \cdot \lim_{x \to 0} \ \frac{1}{\cot (x+\frac{\pi}{3})} \\[8pt] &= (1)^2 \cdot \frac{1}{\cot(0+\frac{\pi}{3})} = 1 \cdot \frac{1}{\cot \frac{\pi}{3}} \\[8pt] &= \tan \frac{\pi}{3} = \sqrt{3} \end{aligned}
Jawaban E.