The values of $\text{Lim}_{x \to 0^+} \frac{f(-x)-x^2}{1-\cos x}$ where $[\cdot]$ denote greatest integer function and $(\cdot)$ denote fraction part function.
Step-by-Step Solution
Key Concept: Use the standard limit $\lim_{x \to 0} \frac{1-\cos x}{x^2} = 1/2$ in combination with function properties to evaluate complex limits.
Evaluate $\lim_{x \to 0^+} \frac{f(-x)x^2}{\frac{1-\cos x}{[f(x)]} \cdot \frac{1-\cos x}{[f(x)]}}$. Given that the numerator involves $f(-x)x^2$ and using $1 - \cos x \sim x^2/2$ as $x \to 0$, combined with properties of $f$, the limit evaluates to $3x^2/(1-\cos x) = 6 \times 2 = 12$.
Correct Answer: 2