Jika \( \displaystyle a = \lim_{x \to 2} \ \frac{x^2-4}{2-\sqrt{x+2}}\), maka nilai \((4-a)\) adalah...
- -20
- -12
- -4
- 12
- 20
(UM UGM 2013)
Pembahasan:
\begin{aligned} a &= \lim_{x \to 2} \ \frac{x^2-4}{2-\sqrt{x+2}} \\[8pt] &= \lim_{x \to 2} \ \frac{x^2-4}{2-\sqrt{x+2}} \times \frac{2+\sqrt{x+2}}{2+\sqrt{x+2}} \\[8pt] &= \lim_{x \to 2} \ \frac{(x-2)(x+2)(2+\sqrt{x+2})}{4-(x+2)} \\[8pt] &= \lim_{x \to 2} \ \frac{(x-2)(x+2)(2+\sqrt{x+2})}{-(x-2)} \\[8pt] &= \lim_{x \to 2} \ -(x+2)(2+\sqrt{x+2}) \\[8pt] &= -(2+2)(2+\sqrt{2+2}) \\[8pt] a &= -16 \\[8pt] \Rightarrow (4-a) &= 4-(-16) = 20 \end{aligned}
Jawaban E.