_ x → (π)/(4) 8 2 - ( x + x) ^ 7 2 - 2 2x is equal to

$\lim_{x \to \dfrac{\pi}{4}}\dfrac{8\sqrt{2} - {\left(\cos x + \sin x\right)}^{7}}{\sqrt{2} - \sqrt{2}\sin 2x}\left(\dfrac{0}{0}form\right)$

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{- 7{\left(\cos x + \sin x\right)}^{6}\left(- \sin x + \cos x\right)}{- 2\sqrt{2}\cos 2x}$ (using L'Hospital Rule)

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{56\left(\cos x - \sin x\right)}{2\sqrt{2}\cos 2x}\left(\dfrac{0}{0}form\right)$

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{- 56\left(\sin x + \cos x\right)}{- 4\sqrt{2}\sin 2x}$ (using L'Hospital Rule)

$= \dfrac{56\sqrt{2}}{4\sqrt{2}} = 14$