`underset(x to 2)(Lt) {[x - 2] + [2 - x] - x} =` where [.] represents greater intergral function

Options

A. 0
B. 3
C. `-3`
D. `-2`

Answer

To solve the limit problem \(\lim_{x \to 2} \left[ \lfloor x - 2 \rfloor + \lfloor 2 - x \rfloor - x \right]\), where \(\lfloor . \rfloor\) represents the greatest integer function, we will evaluate the limit from both the right-hand side (as \(x\) approaches 2 from values greater than 2) and the left-hand side (as \(x\) approaches 2 from values less than 2).

### Step 1: Evaluate the Right-Hand Limit (\(x \to 2^+\))

1. **Substituting \(x = 2^+\)**:
- \(x - 2\) approaches \(0^+\) (a small positive number).
- \(2 - x\) approaches \(0^-\) (a small negative number).

2. **Calculate \(\lfloor x - 2 \rfloor\)**:
- Since \(x - 2\) is a small positive number, \(\lfloor x - 2 \rfloor = 0\).

3. **Calculate \(\lfloor 2 - x \rfloor\)**:
- Since \(2 - x\) is a small negative number, \(\lfloor 2 - x \rfloor = -1\).

4. **Substituting into the limit expression**:
\[ \lfloor x - 2 \rfloor + \lfloor 2 - x \rfloor - x = 0 - 1 - 2 = -3 \]

### Step 2: Evaluate the Left-Hand Limit (\(x \to 2^-\))

1. **Substituting \(x = 2^-\)**:
- \(x - 2\) approaches \(0^-\) (a small negative number).
- \(2 - x\) approaches \(0^+\) (a small positive number).

2. **Calculate \(\lfloor x - 2 \rfloor\)**:
- Since \(x - 2\) is a small negative number, \(\lfloor x - 2 \rfloor = -1\).

3. **Calculate \(\lfloor 2 - x \rfloor\)**:
- Since \(2 - x\) is a small positive number, \(\lfloor 2 - x \rfloor = 0\).

4. **Substituting into the limit expression**:
\[ \lfloor x - 2 \rfloor + \lfloor 2 - x \rfloor - x = -1 + 0 - 2 = -3 \]

### Step 3: Conclusion

Both the right-hand limit and the left-hand limit yield the same value:
\[ \lim_{x \to 2^+} \left[ \lfloor x - 2 \rfloor + \lfloor 2 - x \rfloor - x \right] = -3 \]
\[ \lim_{x \to 2^-} \left[ \lfloor x - 2 \rfloor + \lfloor 2 - x \rfloor - x \right] = -3 \]

Thus, the overall limit is:
\[ \lim_{x \to 2} \left[ \lfloor x - 2 \rfloor + \lfloor 2 - x \rfloor - x \right] = -3 \]

### Final Answer:
\(-3\)