Nilai \( \displaystyle \lim_{x \to 2} \frac{\sin (2x-4)}{2-\sqrt{6-x}} = \cdots \)
- -8
- -2
- 0
- 2
- 8
(SBMPTN 2018)
Pembahasan:
\begin{aligned} \lim_{x \to 2} \ \frac{\sin (2x-4)}{2-\sqrt{6-x}} &= \lim_{x \to 2} \ \frac{\sin (2x-4)}{2-\sqrt{6-x}} \cdot \frac{2+\sqrt{6-x}}{2+\sqrt{6-x}} \\[8pt] &= \lim_{x \to 2} \ \frac{(2+\sqrt{6-x})\sin (2x-4)}{4 - (6-x)} \\[8pt] &= \lim_{x \to 2} \ \frac{(2+\sqrt{6-x})\sin 2(x-2)}{x-2} \\[8pt] &= \lim_{x \to 2} \ (2+\sqrt{6-x}) \cdot \lim_{x \to 2} \ \frac{\sin 2(x-2)}{x-2} \\[8pt] &= (2+\sqrt{6-2}) \cdot 2 \\[8pt] &= (2 + \sqrt{4}) \cdot 2 = 8 \end{aligned}
Jawaban E.