Nilai \( \displaystyle \lim_{x \to 0} \ \frac{x}{2 \csc x \ (1-\sqrt{\cos x})} = \cdots \)
- -2
- -1
- 0
- 1
- 2
Pembahasan:
\begin{aligned} \lim_{x \to 0} \ \frac{x}{2 \csc x \ (1-\sqrt{\cos x})} &= \lim_{x \to 0} \ \frac{x}{2 \csc x \ (1-\sqrt{\cos x})} \cdot \frac{1+\sqrt{\cos x}}{1+\sqrt{\cos x}} \\[8pt] &= \lim_{x \to 0} \ \frac{x \ (1+\sqrt{\cos x})}{\frac{2}{\sin x} \ (1-\cos x)} \\[8pt] &= \lim_{x \to 0} \ \frac{x \sin x \ (1+\sqrt{\cos x})}{2 \ (2 \sin^2 \frac{1}{2}x)} \\[8pt] &= \lim_{x \to 0} \ \frac{x}{4 \sin \frac{1}{2}x} \cdot \lim_{x \to 0} \ \frac{\sin x}{\sin \frac{1}{2}x} \cdot \lim_{x \to 0} \ (1+\sqrt{\cos x}) \\[8pt] &= \frac{1}{4 \cdot \frac{1}{2}} \cdot \frac{1}{\frac{1}{2}} \cdot (1+\sqrt{\cos 0}) \\[8pt] &= \frac{1}{2} \cdot 2 \cdot (1+1) = 2 \end{aligned}
Jawaban E.