Nilai \( \displaystyle \lim_{x\to 1} \ \frac{\sin \left( 1 - \frac{1}{x} \right) \cos \left( 1 - \frac{1}{x} \right)}{x-1} = \cdots \)
Pembahasan:
\begin{aligned} \lim_{x\to 1} \ \frac{\sin \left( 1 - \frac{1}{x} \right) \cos \left( 1 - \frac{1}{x} \right)}{x-1} &= \lim_{x\to 1} \ \frac{\sin \left( \frac{x-1}{x} \right) \cos \left( \frac{x-1}{x} \right)}{x-1} \\[8pt] &= \lim_{x\to 1} \ \frac{\sin \frac{1}{x} (x-1) \cos\frac{1}{x} (x-1)}{x-1} \\[8pt] &= \lim_{x\to 1} \ \frac{\sin \frac{1}{x} (x-1)}{x-1} \cdot \lim_{x\to 1} \ \cos\frac{1}{x} (x-1) \\[8pt] &= \frac{1}{x} \cdot \cos 0 = \frac{1}{x} \cdot 1 \\[8pt] &= \frac{1}{x} \end{aligned}