Evaluate the limit: $\mathop {\text{Limit}}\limits_{x \to 4} \frac{(\cos \alpha)^x - (\sin \alpha)^x - \cos 2\alpha}{x - 4}$,where $0 < \alpha < \frac{\pi}{2}$.
- A$(\cos \alpha)^4 \ln(\cos \alpha) + (\sin \alpha)^4 \ln(\sin \alpha)$
- B$-(\cos \alpha)^4 \ln(\cos \alpha) - (\sin \alpha)^4 \ln(\sin \alpha)$
- C$(\cos \alpha)^4 \ln(\cos \alpha) - (\sin \alpha)^4 \ln(\sin \alpha)$
- D$-(\cos \alpha)^4 \ln(\cos \alpha) + (\sin \alpha)^4 \ln(\sin \alpha)$
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