If `f(x) = {{:([x]+[-x]",",x ne 2),(" "lambda",",x = 2):}` and f is continuous at x = 2, where `[*]` denotes greatest integer function, then `lambda` is

Options

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

Answer

To solve the problem, we need to determine the value of \( \lambda \) such that the function \( f(x) \) is continuous at \( x = 2 \). The function is defined as follows:

\[ f(x) = \begin{cases} [x] + [-x] & \text{if } x \neq 2 \\ \lambda & \text{if } x = 2 \end{cases} \]

where \([x]\) denotes the greatest integer function.

### Step-by-step Solution:

1. **Understanding Continuity**:
A function is continuous at a point \( c \) if:
\[ \lim_{x \to c} f(x) = f(c) \]
In this case, we need to check continuity at \( x = 2 \):
\[ \lim_{x \to 2} f(x) = f(2) = \lambda \]

2. **Finding the Left-Hand Limit (LHL)**:
We calculate the left-hand limit as \( x \) approaches 2 from the left (\( x \to 2^- \)):
\[ \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} ([x] + [-x]) \]
For \( x < 2 \):
- \([x] = 1\) (since the greatest integer less than 2 is 1)
- \([-x] = -2\) (since for \( x < 2 \), \(-x\) is greater than -2)

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

3. **Finding the Right-Hand Limit (RHL)**:
Now, we calculate the right-hand limit as \( x \) approaches 2 from the right (\( x \to 2^+ \)):
\[ \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} ([x] + [-x]) \]
For \( x > 2 \):
- \([x] = 2\) (since the greatest integer less than or equal to \( x \) is 2)
- \([-x] = -3\) (since for \( x > 2 \), \(-x\) is less than -2)

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

4. **Setting the Limits Equal**:
Since both the left-hand limit and the right-hand limit are equal:
\[ \lim_{x \to 2} f(x) = -1 \]
For \( f(x) \) to be continuous at \( x = 2 \), we set:
\[ \lambda = \lim_{x \to 2} f(x) = -1 \]

5. **Conclusion**:
Therefore, the value of \( \lambda \) that makes \( f(x) \) continuous at \( x = 2 \) is:
\[ \lambda = -1 \]

### Final Answer:
\[ \lambda = -1 \]