Nilai \( \displaystyle \lim_{x \to 3} \ \frac{x-3}{3-\sqrt{x+6}} = \cdots \)
- -6
- -3
- 0
- 3
- 6
(UMB PTN 2012)
Pembahasan:
\begin{aligned} \lim_{x \to 3} \ \frac{x-3}{3-\sqrt{x+6}} &= \lim_{x \to 3} \ \frac{x-3}{3-\sqrt{x+6}} \times \frac{3+\sqrt{x+6}}{3+\sqrt{x+6}} \\[8pt] &= \lim_{x \to 3} \ \frac{(x-3)(3+\sqrt{x+6})}{9-(x+6)} \\[8pt] &= \lim_{x \to 3} \ \frac{(x-3)(3+\sqrt{x+6})}{9-(x+6)} \\[8pt] &= \lim_{x \to 3} \ \frac{(x-3)(3+\sqrt{x+6})}{-(x-3)} \\[8pt] &= \lim_{x \to 3} \ \frac{(3+\sqrt{x+6})}{-1} \\[8pt] &= \frac{3+\sqrt{3+6}}{-1} = -6 \end{aligned}
Jawaban A.