Nilai \( \displaystyle \lim_{x \to 3} \ \frac{x^2-x-6}{\sqrt{3x^2-2}-5} = \cdots \)
- 0
- 25/9
- 25/6
- 25/3
- ∞
(UN SMA IPA 2019)
Pembahasan:
\begin{aligned} \lim_{x \to 3} \ \frac{x^2-x-6}{\sqrt{3x^2-2}-5} &= \lim_{x \to 3} \ \frac{x^2-x-6}{\sqrt{3x^2-2}-5} \times \frac{\sqrt{3x^2-2}+5}{\sqrt{3x^2-2}+5} \\[8pt] &= \lim_{x \to 3} \ \frac{(x^2-x-6)(\sqrt{3x^2-2}+5)}{(3x^2-2)-25} \\[8pt] &= \lim_{x \to 3} \ \frac{(x-3)(x+2)(\sqrt{3x^2-2}+5)}{3(x^2-9)} \\[8pt] &= \lim_{x \to 3} \ \frac{(x-3)(x+2)(\sqrt{3x^2-2}+5)}{3(x-3)(x+3)} \\[8pt] &= \lim_{x \to 3} \ \frac{(x+2)(\sqrt{3x^2-2}+5)}{3(x+3)} \\[8pt] &= \frac{(3+2)(\sqrt{3(3)^2-2}+5)}{3(3+3)} \\[8pt] &= \frac{(5)(\sqrt{25}+5)}{18} = \frac{(5)(10)}{18} = \frac{25}{9} \end{aligned}
Jawaban B.