Find each function value and the limit for \( f(x)=\frac{14-8 x^{3}}{5+x^{3}} \). Use - \( \infty \) or \( \infty \) where appropriate. \( \begin{array}{lll}(\text { (A) } f(-10) & \text { (B) } f(-20) & \text { (C) } \lim _{x \rightarrow-\infty} f(x)\end{array} \) \( \begin{array}{ll}\text { (A) } f(-10)=\square & \text { (Round to the nearest thousandih as needed.) }\end{array} \)

Let's first evaluate the function \( f(x)=\frac{14-8 x^{3}}{5+x^{3}} \) for \( f(-10) \) and \( f(-20) \), and then find the limit as \( x \) approaches negative infinity. **(A)** To find \( f(-10) \): \[ f(-10) = \frac{14 - 8(-10)^3}{5 + (-10)^3} = \frac{14 - 8(-1000)}{5 - 1000} = \frac{14 + 8000}{5 - 1000} = \frac{8014}{-995} \approx -8.06 \, \text{(to three decimal places)} \] **(B)** Now let's find \( f(-20) \): \[ f(-20) = \frac{14 - 8(-20)^3}{5 + (-20)^3} = \frac{14 - 8(-8000)}{5 - 8000} = \frac{14 + 64000}{5 - 8000} = \frac{64014}{-7995} \approx -8.00 \, \text{(to three decimal places)} \] **(C)** For the limit as \( x \) approaches negative infinity: As \( x \to -\infty \), the cubic terms dominate: \[ \lim _{x \rightarrow-\infty} f(x) = \lim_{x \to -\infty} \frac{-8 x^{3}}{x^{3}} = \lim_{x \to -\infty} -8 = -8 \] Summing it up: \[ \text{(A) } f(-10) \approx -8.060 \\ \text{(B) } f(-20) \approx -8.000 \\ \text{(C) } \lim _{x \rightarrow-\infty} f(x) = -8 \]