Find Difference in Polynomial Values Given Extrema and Limit Condition

Given that $\lim_{x \to 0} \frac{f(x)}{x^3} = -5$ for a degree $5$ polynomial $f(x)$, we can determine that the lower-degree terms of $f(x)$ are $-5x^3$ and all terms of degree less than 3 are zero, giving $f(x) = ax^5 + bx^4 - 5x^3$.

Taking the derivative $f'(x) = 5ax^4 + 4bx^3 - 15x^2 = x^2(5ax^2 + 4bx - 15)$, the extrema condition at $x = \pm 1$ implies $x = \pm 1$ are roots of $5ax^2 + 4bx - 15 = 0$. Solving for the coefficients yields $b = 0$ and $a = 3$, so $f(x) = 3x^5 - 5x^3$.

Evaluating at $x = 2$ and $x = -2$: $f(2) = 3(2)^5 - 5(2)^3 = 96 - 40 = 56$ $f(-2) = 3(-2)^5 - 5(-2)^3 = -56$

Thus, $f(2) - f(-2) = 56 - (-56) = 112$.

The correct option is D.