Nilai \( \displaystyle \lim_{x \to 0} \ \frac{1-\cos^3 x}{x \tan x} = \cdots \)
- 0
- 1/2
- 3/4
- 3/2
- 3
(UM UGM 2013)
Pembahasan:
\begin{aligned} \lim_{x \to 0} \ \frac{1-\cos^3 x}{x \tan x} &= \lim_{x \to 0} \ \frac{(1-\cos x)(1 + \cos x + \cos^2 x)}{x \tan x} \\[8pt] &= \lim_{x \to 0} \ \frac{2 \sin^2 \frac{1}{2}x \ (1 + \cos x + \cos^2 x)}{x \tan x} \\[8pt] &= \lim_{x \to 0} \ \frac{2 \sin \frac{1}{2}x}{x} \cdot \lim_{x \to 0} \ \frac{\sin \frac{1}{2}x}{\tan x} \cdot \lim_{x \to 0} \ (1 + \cos x + \cos^2 x) \\[8pt] &= \frac{2 \cdot \frac{1}{2}}{1} \cdot \frac{\frac{1}{2}}{1} \cdot (1 + \cos 0 + \cos^2 0) \\[8pt] &= \frac{1}{1} \cdot \frac{1}{2} \cdot (1+1+1) \\[8pt] &= \frac{3}{2} \end{aligned}
Jawaban D.