Question

Exercițiul 3
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Answer 1

$b)\cos x-\sin x=\sqrt{2}\cdot \left(\dfrac{\sqrt 2}{2}\cos x-\dfrac{\sqrt 2}{2}\sin x\right)=\\=\sqrt{2}\cdot \left(\sin\dfrac{\pi}{4}\cos x-\cos\dfrac{\pi}{4}\sin x\right)=-\sqrt 2\cdot \sin\left(x-\dfrac{\pi}{4}\right)\\\displaystyle\lim_{x\to\frac{\pi}{4}}\dfrac{\cos x-\sin x}{x-\frac{\pi}{4}}=\lim_{x\to\frac{\pi}{4}} -\sqrt 2\cdot \dfrac{\sin\left(x-\frac{\pi}{4}\right)}{x-\frac{\pi}{4}}=\boxed{-\sqrt 2}$