Let f(x)=`ltxgt^(**)`, where `lt xgt^(**)` is the distance from x to the integer nearest to x then `lim_(x to 2)` f(x) is

Options

A. 2
B. 1
C. 0
D. none of these

Answer

To solve the problem, we need to find the limit of the function \( f(x) \) as \( x \) approaches 2, where \( f(x) \) is defined as the distance from \( x \) to the nearest integer.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) \) represents the distance from \( x \) to the nearest integer. If \( n \) is the nearest integer to \( x \), then:
\[ f(x) = |x - n| \]

2. **Identifying the Nearest Integer**:
For \( x \) approaching 2, we need to consider two cases:
- When \( x \) is exactly 2 (which is an integer).
- When \( x \) is slightly less than or greater than 2 (i.e., \( x = 2 - h \) or \( x = 2 + h \) where \( h \) is a small positive number).

3. **Case 1: \( x = 2 \)**:
When \( x = 2 \), the nearest integer is 2 itself. Therefore:
\[ f(2) = |2 - 2| = 0 \]

4. **Case 2: \( x \) approaches 2 from the left (\( x = 2 - h \))**:
Here, \( h \) is a small positive number. The nearest integer to \( 2 - h \) is still 2. Thus:
\[ f(2 - h) = |(2 - h) - 2| = |-h| = h \]
As \( h \) approaches 0, \( f(2 - h) \) approaches 0.

5. **Case 3: \( x \) approaches 2 from the right (\( x = 2 + h \))**:
Similarly, for \( x = 2 + h \), the nearest integer is still 2. Therefore:
\[ f(2 + h) = |(2 + h) - 2| = |h| = h \]
As \( h \) approaches 0, \( f(2 + h) \) also approaches 0.

6. **Conclusion**:
Since both the left-hand limit and the right-hand limit as \( x \) approaches 2 are equal to 0, we can conclude:
\[ \lim_{x \to 2} f(x) = 0 \]

### Final Answer:
\[ \lim_{x \to 2} f(x) = 0 \]