Nilai \( \displaystyle \lim_{x \to 1} \ \left( \frac{1}{1-x} - \frac{2}{x-x^3} \right) = \cdots \)
- \( -\frac{3}{2} \)
- \( -\frac{2}{3} \)
- \( \frac{2}{3} \)
- \( 1 \)
- \( \frac{3}{2} \)
(UM UGM 2005)
Pembahasan:
\begin{aligned} \lim_{x \to 1} \ \left( \frac{1}{1-x} - \frac{2}{x-x^3} \right) &= \lim_{x \to 1} \ \left( \frac{1}{1-x} - \frac{2}{(x)(1-x)(1+x)} \right) \\[8pt] &= \lim_{x \to 1} \ \frac{x(x+1)-2}{(x)(1-x)(1+x)} \\[8pt] &= \lim_{x \to 1} \ \frac{x^2+x-2}{(x)(1-x)(1+x)} \\[8pt] &= \lim_{x \to 1} \ \frac{(x-1)(x+2)}{(x)(1-x)(1+x)} \\[8pt] &= \lim_{x \to 1} \ \frac{-(1-x)(x+2)}{(x)(1-x)(1+x)} \\[8pt] &= \lim_{x \to 1} \ \frac{-(x+2)}{(x)(1+x)} = \frac{-(1+2)}{(1)(1+1)} \\[8pt] &= -\frac{3}{2} \end{aligned}
Jawaban A.