To solve the limit [tex]\(\lim _{x \rightarrow \frac{-\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^7}{\sqrt{2}-\sqrt{2} \sin 2 x}\)[/tex], let's analyze the given expression step-by-step:
1. Expression Setup:
[tex]\[\lim_{x \to -\frac{\pi}{4}} \frac{8\sqrt{2} - (\cos x + \sin x)^7}{\sqrt{2} - \sqrt{2}\sin(2x)}\][/tex]
2. Substitute [tex]\( x = -\frac{\pi}{4} \)[/tex] where possible:
- Evaluate inside the trigonometric functions:
[tex]\[
\cos\left(-\frac{\pi}{4}\right) = \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\][/tex]
Therefore,
[tex]\[
\cos\left(-\frac{\pi}{4}\right) + \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = -\sqrt{2}
\][/tex]
- Simplify the numerator:
[tex]\[
(\cos x + \sin x)^7 = (-\sqrt{2})^7 = -(\sqrt{2})^7
\][/tex]
Now,
[tex]\[
8\sqrt{2} - (-(\sqrt{2})^7) = 8\sqrt{2} + (\sqrt{2})^7 = 8\sqrt{2} + 4\sqrt{2}
\][/tex]
Simplifying,
[tex]\[
8\sqrt{2} + 4\sqrt{2} = 12\sqrt{2}
\][/tex]
- Evaluate [tex]\( \sin(2x) \)[/tex]:
[tex]\[
2x = 2 \left(-\frac{\pi}{4}\right) = -\frac{\pi}{2}
\][/tex]
[tex]\[
\sin\left(-\frac{\pi}{2}\right) = -1
\][/tex]
Therefore,
[tex]\[
\sqrt{2} - \sqrt{2}\sin(2x) = \sqrt{2} - \sqrt{2}(-1) = \sqrt{2} + \sqrt{2} = 2\sqrt{2}
\][/tex]
3. Form the limit expression with [tex]\( x = -\frac{\pi}{4} \)[/tex]:
[tex]\[
\frac{12\sqrt{2}}{2\sqrt{2}} = \frac{12}{2} = 6
\][/tex]
4. Verify consistency:
Both numerator and denominator properly simplify, indicating a straightforward limit process without indeterminate forms.
The final simplified value is:
[tex]\[
\boxed{4}
\][/tex]
