Nilai \( \displaystyle \lim_{x \to \frac{\pi}{4}} \ \frac{1 - \tan x}{\sin x - \cos x} = \cdots \)
Pembahasan:
\begin{aligned} \lim_{x \to \frac{\pi}{4}} \ \frac{1 - \tan x}{\sin x - \cos x} &= \lim_{x \to \frac{\pi}{4}} \ \frac{1 - \frac{\sin x}{\cos x}}{\sin x - \cos x} \\[8pt] &= \lim_{x \to \frac{\pi}{4}} \ \frac{ \displaystyle \frac{\cos x - \sin x}{\cos x}}{-(\cos x-\sin x)} \\[8pt] &= \lim_{x \to \frac{\pi}{4}} \ \frac{\cos x-\sin x}{-\cos x (\cos x-\sin x)} \\[8pt] &= - \lim_{x \to \frac{\pi}{4}} \ \frac{1}{\cos x} = - \frac{1}{\cos \frac{\pi}{4}} \\[8pt] &= -\frac{1}{\frac{1}{2}\sqrt{2}} = -\sqrt{2} \end{aligned}