Nilai \( \displaystyle \lim_{x \to 2} \ \left( \frac{6}{x^2-x-2} - \frac{2}{x-2} \right) = \cdots \)
- \( -1 \)
- \( -\frac{2}{3} \)
- \( -\frac{1}{3} \)
- \( \frac{1}{3} \)
- \( \frac{2}{3} \)
(UM UGM 2010)
Pembahasan:
\begin{aligned} \lim_{x \to 2} \ \left( \frac{6}{x^2-x-2} - \frac{2}{x-2} \right) &= \lim_{x \to 2} \ \left( \frac{6}{(x-2)(x+1)} - \frac{2}{x-2} \right) \\[8pt] &= \lim_{x \to 2} \ \left( \frac{6}{(x-2)(x+1)} - \frac{2(x+1)}{(x-2)(x+1)} \right) \\[8pt] &= \lim_{x \to 2} \ \frac{6-2(x+1)}{(x-2)(x+1)} = \lim_{x \to 2} \ \frac{4-2x}{(x-2)(x+1)} \\[8pt] &= \lim_{x \to 2} \ \frac{-2(x-2)}{(x-2)(x+1)} = \lim_{x \to 2} \ \frac{-2}{(x+1)} \\[8pt] &= \frac{-2}{2+1} = -\frac{2}{3} \end{aligned}
Jawaban B.