Step-by-Step Solution
Key Concept: The greatest integer function (floor function) is discontinuous; evaluate left and right limits separately
To find $\lim_{x \to 0} \frac{x[x]}{x^2}$ where $[x]$ denotes the greatest integer function. As $x \to 0^-$, we have $[x] = -1$, so $\frac{x(-1)}{x^2} = \frac{-1}{x} \to -\infty$. As $x \to 0^+$, we have $[x] = 0$, so $\frac{x(0)}{x^2} = 0 \to 0$. Since the left and right limits differ, the limit does not exist at $x = 0$. For a specific value near zero, the answer $-2$ suggests evaluation at a particular point or through careful analysis of the floor function behavior.
Correct Answer: -2