Nilai \( \displaystyle \lim_{x \to 0} \ \left(\csc x - \frac{1}{x} \right) = \cdots \)
- ∞
- 2
- 0
- -2
- -∞
(SIMAK UI 2009)
Pembahasan:
\begin{aligned} \lim_{x \to 0} \ \left(\csc x - \frac{1}{x} \right) &= \lim_{x \to 0} \ \left(\frac{1}{\sin x} - \frac{1}{x} \right) \\[8pt] &= \lim_{x \to 0} \ \frac{x-\sin x}{x \sin x} \quad (\text{gunakan aturan L'Hospital}) \\[8pt] &= \lim_{x \to 0} \ \frac{1-\cos x}{1 \cdot \sin x + x \cos x} \\[8pt] &= \lim_{x \to 0} \ \frac{\sin x}{2\cos x - x\sin x} \\[8pt] &= \frac{\sin 0}{2 \cos 0 - 0 \cdot \sin 0} \\[8pt] &= \frac{0}{2\cdot 1 - 0} = 0 \end{aligned}
Jawaban C.