If `lim _(x to c ^(-)) {ln x} and lim _(xto c ^(+)) {ln x}` exist finitely but they are not equal (where {.} denotes fractional part function), then:

Options

A. c' can take only rational values
B. c' can take only irrational values
C. c' can take infinite values in which only one is irrational
D. c' can take infinite values I which only one is rational

Answer

To solve the problem, we need to analyze the limits of the fractional part of the natural logarithm function as \( x \) approaches \( c \) from the left and right. The fractional part function is denoted as \( \{x\} = x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \).

### Step-by-Step Solution:

1. **Understanding the Limits**:
We need to evaluate \( \lim_{x \to c^-} \{\ln x\} \) and \( \lim_{x \to c^+} \{\ln x\} \). We know that these limits exist finitely but are not equal.

2. **Choosing a Value for \( c \)**:
Let's consider \( c = e \) (the base of the natural logarithm). As \( x \) approaches \( e \) from the left (\( c^- \)):
\[ \ln x \to \ln e = 1 \quad \text{(from the left)} \]
Thus, \( \lim_{x \to e^-} \{\ln x\} = \{1\} = 1 - 0 = 1 \).

3. **Evaluating the Right Limit**:
As \( x \) approaches \( e \) from the right (\( c^+ \)):
\[ \ln x \to \ln e = 1 \quad \text{(from the right)} \]
Thus, \( \lim_{x \to e^+} \{\ln x\} = \{1\} = 0 \quad \text{(since we take the fractional part)} \).

4. **Conclusion from the Limits**:
We find that:
\[ \lim_{x \to e^-} \{\ln x\} = 1 \quad \text{and} \quad \lim_{x \to e^+} \{\ln x\} = 0 \]
Since these two limits exist but are not equal, we confirm the conditions of the problem.

5. **Generalizing the Result**:
This behavior can be generalized for other values of \( c \) that are positive and irrational. For rational values of \( c \), the limits will also differ, but the key takeaway is that for irrational values of \( c \), the limits will yield fractional parts that differ.

### Final Answer:
The limits exist finitely but are not equal, indicating that \( c \) must be an irrational number.