Nilai \( \displaystyle \lim_{x \to 1} \frac{\sin 2(x-1)}{(x^2-2x+1) \cot \frac{1}{2}(x-1)} = \cdots \)
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(SIMAK UI 2012)
Pembahasan:
\begin{aligned} \lim_{x \to 1} \ \frac{\sin 2(x-1)}{(x^2-2x+1) \cot \frac{1}{2}(x-1)} &= \lim_{x \to 1} \ \frac{\sin 2(x-1)}{(x-1)(x-1) \ \frac{\cos \frac{1}{2}(x-1) }{\sin \frac{1}{2}(x-1)} } \\[8pt] &= \lim_{x \to 1} \ \frac{\sin 2(x-1) \sin \frac{1}{2}(x-1)}{(x-1)(x-1) \ \cos \frac{1}{2}(x-1) } \\[8pt] &= \lim_{x \to 1} \ \frac{\sin 2(x-1)}{(x-1)} \cdot \lim_{x \to 1} \ \frac{\sin \frac{1}{2}(x-1)}{(x-1)} \cdot \lim_{x \to 1} \ \frac{1}{\cos \frac{1}{2}(x-1) } \\[8pt] &= 2 \cdot \frac{1}{2} \cdot \frac{1}{\cos 0} = 2 \cdot \frac{1}{2} \cdot \frac{1}{1} = 1 \end{aligned}
Jawaban C.