Answer is: option1
\(-\frac{\pi}{8}\)Solution:
Substituting \( w = -4 \):
\[ \sin(\pi(-4)) = \sin(-4\pi) = \sin(4\pi) = 0 \]
\[ (-4)^2 - 16 = 16 - 16 = 0 \]
Since we get \( \frac{0}{0} \), we apply **L'Hôpital's Rule**, which states:
\[ \lim_{w \to c} \frac{f(w)}{g(w)} = \lim_{w \to c} \frac{f'(w)}{g'(w)} \]
Numerator:
\[ \frac{d}{dw} \sin(\pi w) = \pi \cos(\pi w) \]
Denominator:
\[ \frac{d}{dw} (w^2 - 16) = 2w \]
Substituting \( w = -4 \):
\[ \cos(\pi(-4)) = \cos(-4\pi) = \cos(4\pi) = 1 \]
\[ 2(-4) = -8 \]
\[ \frac{\pi \cdot 1}{-8} = -\frac{\pi}{8} \]
Final Answer: \( -\frac{\pi}{8} \)