If a and b are positive integers then

Options

A. `lim_( x to 0+) x/a [b/x]=a/b`
B. `lim_( x to 0+) x/a [b/x]=ab`
C. `lim_(x to 0+) a/x [x/b]=b/a`
D. `lim_( x to 0+) x/a [b/x]=b/a`

Answer

To solve the given limit problem, we need to evaluate the limit:

\[ \lim_{x \to 0^+} \frac{x}{a} \left\lfloor \frac{b}{x} \right\rfloor \]

where \( a \) and \( b \) are positive integers.

### Step-by-Step Solution:

1. **Understanding the Floor Function**:
The floor function \( \left\lfloor \frac{b}{x} \right\rfloor \) gives the greatest integer less than or equal to \( \frac{b}{x} \). As \( x \) approaches \( 0^+ \), \( \frac{b}{x} \) approaches \( +\infty \), and thus \( \left\lfloor \frac{b}{x} \right\rfloor \) also approaches \( +\infty \).

2. **Rewriting the Limit**:
We can express the limit as:
\[ \lim_{x \to 0^+} \frac{x}{a} \left\lfloor \frac{b}{x} \right\rfloor = \lim_{x \to 0^+} \frac{x}{a} \left( \frac{b}{x} - \left\{ \frac{b}{x} \right\} \right) \]
where \( \left\{ \frac{b}{x} \right\} \) is the fractional part of \( \frac{b}{x} \).

3. **Simplifying the Expression**:
This can be simplified to:
\[ \lim_{x \to 0^+} \left( \frac{b}{a} - \frac{x}{a} \left\{ \frac{b}{x} \right\} \right) \]

4. **Evaluating the Limit**:
As \( x \to 0^+ \), \( \left\{ \frac{b}{x} \right\} \) remains bounded between \( 0 \) and \( 1 \). Therefore, \( \frac{x}{a} \left\{ \frac{b}{x} \right\} \) approaches \( 0 \) because \( x \) approaches \( 0 \). Thus, we have:
\[ \lim_{x \to 0^+} \frac{x}{a} \left\{ \frac{b}{x} \right\} = 0 \]

5. **Final Result**:
Therefore, the limit simplifies to:
\[ \lim_{x \to 0^+} \left( \frac{b}{a} - 0 \right) = \frac{b}{a} \]

### Conclusion:
The value of the limit is:
\[ \frac{b}{a} \]