To find the limit of [tex]\(\lim_{x \to \frac{\pi}{2}} \frac{8 \sqrt{2} - (\cos x + \sin x)^7}{\sqrt{2} - \sqrt{2} \sin 2x}\)[/tex], we can follow these steps:
1. Evaluate the behavior of the numerator as [tex]\( x \)[/tex] approaches [tex]\( \frac{\pi}{2} \)[/tex]:
When [tex]\( x \)[/tex] approaches [tex]\( \frac{\pi}{2} \)[/tex]:
- [tex]\(\cos\left(\frac{\pi}{2}\right) = 0\)[/tex]
- [tex]\(\sin\left(\frac{\pi}{2}\right) = 1\)[/tex]
Therefore,
[tex]\[
\cos x + \sin x \rightarrow 0 + 1 = 1
\][/tex]
and consequently,
[tex]\[
(\cos x + \sin x)^7 \rightarrow 1^7 = 1.
\][/tex]
Plugging this into the numerator:
[tex]\[
8\sqrt{2} - (\cos x + \sin x)^7 \rightarrow 8\sqrt{2} - 1.
\][/tex]
2. Evaluate the behavior of the denominator as [tex]\( x \)[/tex] approaches [tex]\( \frac{\pi}{2} \)[/tex]:
For the denominator, we note that:
- [tex]\( \sin(2x) = \sin\left(2 \cdot \frac{\pi}{2}\right) = \sin(\pi) = 0 \)[/tex].
Plugging this into the denominator:
[tex]\[
\sqrt{2} - \sqrt{2} \sin(2x) \rightarrow \sqrt{2} - \sqrt{2} \cdot 0 = \sqrt{2}.
\][/tex]
3. Combine and simplify:
Our expression now simplifies to:
[tex]\[
\frac{8 \sqrt{2} - 1}{\sqrt{2}}.
\][/tex]
Let's perform the division:
[tex]\[
\frac{8 \sqrt{2} - 1}{\sqrt{2}} = \frac{8 \sqrt{2}}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 8 - \frac{1}{\sqrt{2}}.
\][/tex]
4. Further simplification:
We can simplify [tex]\(\frac{1}{\sqrt{2}}\)[/tex] by rationalizing the denominator:
[tex]\[
\frac{1}{\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}.
\][/tex]
Thus, the expression becomes:
[tex]\[
8 - \frac{\sqrt{2}}{2} = 8 - 0.707 \approx 7.292.
\][/tex]
5. Conclusion:
Therefore, the limit is:
[tex]\[
\boxed{7.292893218813452}.
\][/tex]
