if `[x]` denotes the greatest integer less than or equal to x, than `lim_(xrarr0)(x[x])/(sin|x|)`, is

Options

A. 0
B. 1
C. non-existant
D. none of these

Explanation

To solve the limit \( \lim_{x \to 0} \frac{x[x]}{\sin |x|} \), where \([x]\) denotes the greatest integer less than or equal to \(x\), we will analyze the limit from both sides: as \(x\) approaches \(0\) from the left (\(0^-\)) and from the right (\(0^+\)).

### Step-by-Step Solution:

1. **Evaluate the limit as \(x\) approaches \(0\) from the left (\(0^-\)):**
- For \(x < 0\), the greatest integer function \([x]\) will be \(-1\) because it is the largest integer less than \(x\).
- Therefore, we can rewrite the limit:
\[ \lim_{x \to 0^-} \frac{x[x]}{\sin |x|} = \lim_{x \to 0^-} \frac{x(-1)}{\sin(-x)} = \lim_{x \to 0^-} \frac{-x}{-\sin x} = \lim_{x \to 0^-} \frac{x}{\sin x} \]
- As \(x\) approaches \(0\), we know that \(\frac{x}{\sin x} \to 1\).
- Thus,
\[ \lim_{x \to 0^-} \frac{x[x]}{\sin |x|} = 1. \]

2. **Evaluate the limit as \(x\) approaches \(0\) from the right (\(0^+\)):**
- For \(x > 0\), the greatest integer function \([x]\) will be \(0\) because it is the largest integer less than or equal to \(x\).
- Therefore, we can rewrite the limit:
\[ \lim_{x \to 0^+} \frac{x[x]}{\sin |x|} = \lim_{x \to 0^+} \frac{x(0)}{\sin x} = \lim_{x \to 0^+} \frac{0}{\sin x} = 0. \]

3. **Compare the two limits:**
- From the left, we found:
\[ \lim_{x \to 0^-} \frac{x[x]}{\sin |x|} = 1. \]
- From the right, we found:
\[ \lim_{x \to 0^+} \frac{x[x]}{\sin |x|} = 0. \]
- Since the two one-sided limits are not equal, the overall limit does not exist.

### Conclusion:
Thus, the limit \( \lim_{x \to 0} \frac{x[x]}{\sin |x|} \) does not exist.