Nilai \( \displaystyle \lim_{x \to y} \ \frac{\sin x - \sin y}{x-y} = \cdots \)
- \( \sin x \)
- \( \sin y \)
- 0
- \( \cos x \)
- \( \cos y \)
(UM UNDIP 2010)
Pembahasan:
\begin{aligned} \lim_{x \to y} \ \frac{\sin x - \sin y}{x-y} &= \lim_{x \to y} \ \frac{2 \cos \frac{1}{2} (x+y) \sin \frac{1}{2} (x-y) }{x-y} \\[8pt] &= \lim_{x \to y} \ \frac{\sin \frac{1}{2} (x-y) }{x-y} \cdot \lim_{x \to y} \ 2 \cos \frac{1}{2} (x+y) \\[8pt] &= \frac{1}{2} \cdot 2 \cos \frac{1}{2}(2y) \\[8pt] &= \cos y \end{aligned}
Jawaban E.