$\underset{x\to \frac{\pi }{4}}{\lim }\frac{8\sqrt{2}-{\left(\cos x+\sin x\right)}^7}{\sqrt{2}-\sqrt{2}\sin 2x} \left(\frac{0}{0}\mathrm{form}\right)$

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

$=\underset{x\to \frac{\pi }{4}}{\lim }\frac{56\left(\cos x-\sin x\right)}{2\sqrt{2}\cos 2x} \left(\frac{0}{0} \mathrm{form}\right)$

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

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