If f(x) = { `sin[x]/([x]),[x] != 0 ; 0, [x] = 0}` , Where[.] denotes the greatest integer function, then `lim_(x rarr 0) f(x)` is equal to

Options

A. 1
B. 0
C. `-1`
D. Does not exist

Explanation

To solve the limit problem given the function \( f(x) = \frac{\sin[\cdot]}{[\cdot]} \) where \( [\cdot] \) denotes the greatest integer function, we need to find \( \lim_{x \to 0} f(x) \).

### Step-by-Step Solution:

1. **Understanding the Function**:
- The function is defined as:
\[ f(x) = \begin{cases} \frac{\sin([\cdot])}{[\cdot]} & \text{if } [x] \neq 0 \\ 0 & \text{if } [x] = 0 \end{cases} \]
- The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \).

2. **Finding Left-Hand Limit**:
- We need to evaluate \( \lim_{x \to 0^-} f(x) \).
- When \( x \) is approaching \( 0 \) from the left (i.e., \( x < 0 \)), \( [x] = -1 \).
- Therefore, for \( x < 0 \):
\[ f(x) = \frac{\sin(-1)}{-1} = \frac{-\sin(1)}{-1} = \sin(1) \]
- Thus, the left-hand limit is:
\[ \lim_{x \to 0^-} f(x) = \sin(1) \]

3. **Finding Right-Hand Limit**:
- Now, we evaluate \( \lim_{x \to 0^+} f(x) \).
- When \( x \) is approaching \( 0 \) from the right (i.e., \( x > 0 \)), \( [x] = 0 \).
- Therefore, for \( x > 0 \):
\[ f(x) = 0 \]
- Thus, the right-hand limit is:
\[ \lim_{x \to 0^+} f(x) = 0 \]

4. **Comparing the Limits**:
- We have:
\[ \lim_{x \to 0^-} f(x) = \sin(1) \quad \text{and} \quad \lim_{x \to 0^+} f(x) = 0 \]
- Since the left-hand limit \( \sin(1) \) is not equal to the right-hand limit \( 0 \), the overall limit does not exist.

5. **Conclusion**:
- Therefore, we conclude that:
\[ \lim_{x \to 0} f(x) \text{ does not exist.} \]