Limit x mendekati π/4 : 2 cos 2x per cos x - sin x

Limit x mendekati π/4 : 2 cos 2x per cos x – sin x

Jawaban Terkonfirmasi

Ingat identitas:
$boxed{cos 2x=cos^2x-sin^2x}$
Maka,
$$begin{align}lim_{xtofrac{pi}{4}}frac{2cos2x}{cos x-sin x}&=lim_{xtofrac{pi}{4}}frac{2(cos^2x-sin^2x)}{cos x-sin x} \ &=lim_{xtofrac{pi}{4}}frac{2(cos x+sin x)(cos x-sin x)}{cos x-sin x} \ &=lim_{xtofrac{pi}{4}}2(cos x+sin x) \ &=2(cosfrac{pi}{4}+sinfrac{pi}{4}) \ &=2left[frac{sqrt{2}}{2}+frac{sqrt{2}}{2}right] \ &=2timesfrac{2sqrt{2}}{2} \ &=2sqrt{2} end{align}$