Nilai \( \displaystyle \lim_{x \to 0} \ \left( \frac{1}{x} - \frac{1}{x \cos x} \right ) = \cdots \)
- -1
- -1/2
- 0
- 1/2
- 1
(UM UGM 2006)
Pembahasan:
\begin{aligned} \lim_{x \to 0} \ \left( \frac{1}{x} - \frac{1}{x \cos x} \right ) &= \lim_{x \to 0} \ \frac{x \cos x - x}{x^2 \cos x} \\[8pt] &= \lim_{x \to 0} \ \frac{\cos x - 1}{x \cos x} \\[8pt] &= \lim_{x \to 0} \ \frac{-2\sin^2 \frac{1}{2}x}{x \cos 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}{\cos x} \\[8pt] &= -2 \cdot \frac{1}{2} \cdot \frac{\sin 0}{\cos 0} = -1 \cdot \frac{0}{1} = 0 \end{aligned}
Jawaban C.