Nilai \( \displaystyle f(x)= \frac{1-x}{2-\sqrt{x^2+3}} \), maka \( \displaystyle \lim_{x \to 1} \ f(x) = \cdots \)
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(UM UGM 2006)
Pembahasan:
\begin{aligned} \lim_{x \to 1} \ \frac{1-x}{2-\sqrt{x^2+3}} &= \lim_{x \to 1} \ \frac{1-x}{2-\sqrt{x^2+3}} \times \frac{2+\sqrt{x^2+3}}{2+\sqrt{x^2+3}} \\[8pt] &= \lim_{x \to 1} \ \frac{(1-x)(2+\sqrt{x^2+3})}{4-(x^2+3)} \\[8pt] &= \lim_{x \to 1} \ \frac{(1-x)(2+\sqrt{x^2+3})}{1-x^2} \\[8pt] &= \lim_{x \to 1} \ \frac{(1-x)(2+\sqrt{x^2+3})}{(1-x)(1+x)} \\[8pt] &= \lim_{x \to 1} \ \frac{(2+\sqrt{x^2+3})}{(1+x)} = \frac{2+\sqrt{1^2+3}}{1+1} \\[8pt] &= \frac{2+2}{2} = 2 \end{aligned}
Jawaban D.