Nilai \( \displaystyle \lim_{x\to 0} \ \frac{x \cos 2x}{\tan x - \sin 2x} = \cdots \)
- -1
- -1/2
- 0
- 1/2
- 1
Pembahasan:
\begin{aligned} \lim_{x\to 0} \ \frac{x \cos 2x}{\tan x - \sin 2x} &= \lim_{x\to 0} \ \frac{x \cos 2x}{\tan x - \sin 2x} \times \frac{\frac{1}{x}}{\frac{1}{x}} \\[8pt] &= \lim_{x\to 0} \ \frac{\cos 2x}{\frac{\tan x}{x} - \frac{\sin 2x}{x}} \\[8pt] &= \frac{ \displaystyle \lim_{x\to 0} \ \cos 2x }{ \displaystyle \lim_{x\to 0} \ \frac{\tan x}{x} - \lim_{x\to 0} \ \frac{\sin 2x}{x} } \\[8pt] &= \frac{\cos 0}{1 - 2} = \frac{1}{-1} = -1 \end{aligned}
Jawaban A.