Nilai \( \displaystyle \lim_{x \to 1} \ \frac{(x^2+x-2) \sin (x-1)}{x^2 - 2x + 1} = \cdots \)
- 4
- 3
- 0
- \( -\frac{1}{4} \)
- \( -\frac{1}{2} \)
(SPMB 2005)
Pembahasan:
\begin{aligned} \lim_{x \to 1} \ \frac{(x^2+x-2) \sin (x-1)}{x^2 - 2x + 1} &= \lim_{x \to 1} \ \frac{(x+2)(x-1) \sin(x-1)}{(x-1)(x-1)} \\[8pt] &= \lim_{x \to 1} \ \frac{(x+2)\sin(x-1)}{(x-1)} \\[8pt] &= \lim_{x \to 1} \ (x+2) \cdot \lim_{x \to 1} \ \frac{\sin(x-1)}{(x-1)} \\[8pt] &= (1+2) \cdot 1 \\[8pt] &= 3 \end{aligned}
Jawaban B.