Nilai \( \displaystyle \lim_{x \to 3} \ \frac{(x+6) \tan(2x-6)}{(x^2-x-6)} = \cdots \)
- \( -\frac{18}{5} \)
- \( -\frac{9}{5} \)
- \( \frac{9}{5} \)
- \( \frac{18}{5} \)
- \( \frac{27}{5} \)
(UM UGM 2016)
Pembahasan:
\begin{aligned} \lim_{x \to 3} \ \frac{(x+6) \tan(2x-6)}{(x^2-x-6)} &= \lim_{x \to 3} \ \frac{(x+6) \tan 2(x-3)}{(x+2)(x-3)} \\[8pt] &= \lim_{x \to 3} \ \frac{(x+6)}{(x+2)} \cdot \lim_{x \to 3} \ \frac{\tan 2(x-3)}{(x-3)} \\[8pt] &= \frac{(3+6)}{(3+2)} \cdot 2 = \frac{9}{5} \cdot 2 = \frac{18}{5} \end{aligned}
Jawaban D.