Consider the following for the next two items that follow:
Let `f(x)=[x]`, where [.] is the greatest integer function and `g(x)=sinx` be two real valued function over R.
Which one of the following statements is correct?

Options

A. `underset(xrarr0)(lim)(fog)(x)` exists
B. `underset(xrarr0)(lim)(gof)(x)` exists
C. `underset(xrarr0^(-))(lim)(fog)(x)=underset(xrarr0^(-))(lim)(gof)(x)`
D. `underset(xrarr0^(+))(lim)(fog)(x)=underset(xrarr0^(+))(lim)(gof)(x)`

Answer

To solve the problem, we need to analyze the functions \( f(x) = [x] \) (the greatest integer function) and \( g(x) = \sin x \). We will evaluate the limits as \( x \) approaches 0 for \( f(g(x)) \) and \( g(f(x)) \) and check the validity of the statements provided.

### Step-by-step Solution:

1. **Evaluate \( f(g(x)) \)**:
- We have \( g(x) = \sin x \).
- Thus, \( f(g(x)) = f(\sin x) = [\sin x] \).
- We need to find the limit as \( x \) approaches 0:
\[ \lim_{x \to 0} f(g(x)) = \lim_{x \to 0} [\sin x] \]

2. **Calculate Left-Hand Limit**:
- As \( x \) approaches 0 from the left (\( x \to 0^- \)), \( \sin x \) approaches 0 but is slightly negative:
\[ \sin(0^-) \approx -0.0000\ldots \]
- Therefore, \( [\sin(0^-)] = [-0.0000\ldots] = -1 \).

3. **Calculate Right-Hand Limit**:
- As \( x \) approaches 0 from the right (\( x \to 0^+ \)), \( \sin x \) approaches 0 but is slightly positive:
\[ \sin(0^+) \approx 0.0000\ldots \]
- Therefore, \( [\sin(0^+)] = [0.0000\ldots] = 0 \).

4. **Conclusion for \( f(g(x)) \)**:
- The left-hand limit is -1 and the right-hand limit is 0:
\[ \lim_{x \to 0^-} f(g(x)) \neq \lim_{x \to 0^+} f(g(x)) \]
- Thus, \( \lim_{x \to 0} f(g(x)) \) does not exist.

5. **Evaluate \( g(f(x)) \)**:
- We have \( f(x) = [x] \).
- Thus, \( g(f(x)) = g([x]) = \sin([x]) \).
- We need to find the limit as \( x \) approaches 0:
\[ \lim_{x \to 0} g(f(x)) = \lim_{x \to 0} \sin([x]) \]

6. **Calculate Left-Hand Limit**:
- As \( x \) approaches 0 from the left (\( x \to 0^- \)), \( [x] = -1 \):
\[ \sin([-1]) = \sin(-1) \]

7. **Calculate Right-Hand Limit**:
- As \( x \) approaches 0 from the right (\( x \to 0^+ \)), \( [x] = 0 \):
\[ \sin([0]) = \sin(0) = 0 \]

8. **Conclusion for \( g(f(x)) \)**:
- The left-hand limit is \( \sin(-1) \) and the right-hand limit is 0:
\[ \lim_{x \to 0^-} g(f(x)) \neq \lim_{x \to 0^+} g(f(x)) \]
- Thus, \( \lim_{x \to 0} g(f(x)) \) does not exist.

9. **Evaluate \( \lim_{x \to 0} f(g(x)) \) and \( g(f(x)) \)**:
- Since both limits do not exist, we can check the statements provided in the question.

10. **Final Evaluation of Statements**:
- **Statement A**: Limit \( x \to 0 \) of \( f(g(x)) \) exists - **False**.
- **Statement B**: Limit \( x \to 0 \) of \( g(f(x)) \) exists - **False**.
- **Statement C**: \( \lim_{x \to 0^-} f(g(x)) = g(f(x)) \) - **False**.
- **Statement D**: \( \lim_{x \to 0^+} f(g(x)) = g(f(x)) \) - **True** (both equal to 0).

### Conclusion:
The correct statement is **Option D**.