`f(x)={{:([x]+[-x]",", " when "xne2),(k","," when "x=2):}`
If f(x) is continuous at x = 2, then value of `k` will be (where `[*]` denotes G.I.F.)

Options

A. `-1`
B. 1
C. 0
D. 2

Answer

To find the value of \( k \) such that the function \( f(x) \) is continuous at \( x = 2 \), we need to ensure that the limit of \( f(x) \) as \( x \) approaches 2 from both sides equals \( f(2) \).

### Step-by-Step Solution:

1. **Define the Function**:
\[ f(x) = \begin{cases} [x] + [-x] & \text{when } x \neq 2 \\ k & \text{when } x = 2 \end{cases} \]
Here, \( [x] \) denotes the greatest integer function (G.I.F.).

2. **Check Continuity Condition**:
For \( f(x) \) to be continuous at \( x = 2 \), we need:
\[ \lim_{x \to 2} f(x) = f(2) = k \]

3. **Calculate the Left-Hand Limit**:
\[ \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} ([x] + [-x]) \]
As \( x \) approaches 2 from the left (i.e., values like 1.9, 1.99), we have:
- \( [x] = 1 \) (since the greatest integer less than 2 is 1)
- \( [-x] = -2 \) (since the greatest integer less than or equal to -1.9 is -2)

Therefore:
\[ \lim_{x \to 2^-} f(x) = 1 + (-2) = -1 \]

4. **Calculate the Right-Hand Limit**:
\[ \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} ([x] + [-x]) \]
As \( x \) approaches 2 from the right (i.e., values like 2.1, 2.01), we have:
- \( [x] = 2 \) (since the greatest integer less than 3 is 2)
- \( [-x] = -2 \) (since the greatest integer less than or equal to -2.1 is -3)

Therefore:
\[ \lim_{x \to 2^+} f(x) = 2 + (-3) = -1 \]

5. **Set the Limits Equal**:
Since both limits are equal:
\[ \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = -1 \]
We can now set this equal to \( k \):
\[ k = -1 \]

### Conclusion:
Thus, the value of \( k \) for which \( f(x) \) is continuous at \( x = 2 \) is:
\[ \boxed{-1} \]