If `f(x)={{:(x+(1)/(2)",",xlt0),(2x+(3)/(4)",",xge0):},"then"[lim_(xrarr0) f(x)]="(where[.] denotes the greatest integer function)"`

Options

A. `(1)/(2)`
B. `(3)/(4)`
C. does not exist
D. none of these

Explanation

To solve the problem, we need to find the limit of the function \( f(x) \) as \( x \) approaches 0 from both the left and the right, and then determine the greatest integer of that limit.

### Step-by-step Solution:

1. **Identify the function**:
The function is defined as:
\[ f(x) = \begin{cases} x + \frac{1}{2} & \text{if } x < 0 \\ 2x + \frac{3}{4} & \text{if } x \geq 0 \end{cases} \]

2. **Calculate the left-hand limit (LHL)**:
We need to find:
\[ \lim_{x \to 0^-} f(x) \]
As \( x \) approaches 0 from the left, we use the first part of the function:
\[ f(x) = x + \frac{1}{2} \]
Therefore:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \left( x + \frac{1}{2} \right) = 0 + \frac{1}{2} = \frac{1}{2} \]

3. **Calculate the right-hand limit (RHL)**:
Now we find:
\[ \lim_{x \to 0^+} f(x) \]
As \( x \) approaches 0 from the right, we use the second part of the function:
\[ f(x) = 2x + \frac{3}{4} \]
Therefore:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \left( 2x + \frac{3}{4} \right) = 2(0) + \frac{3}{4} = \frac{3}{4} \]

4. **Determine if the limit exists**:
We have:
- LHL = \( \frac{1}{2} \)
- RHL = \( \frac{3}{4} \)

Since LHL \( \neq \) RHL, the limit does not exist:
\[ \lim_{x \to 0} f(x) \text{ does not exist.} \]

5. **Conclusion**:
Since the limit does not exist, we conclude that the answer to the question is:
\[ \text{The limit does not exist.} \]

### Final Answer:
The greatest integer function of the limit is:
\[ \text{Answer: Does not exist.} \]