If a and b are positive and [x] denotes greatest integer less than or equal to x, then find `lim_(xto0^(+)) x/a[(b)/(x)].`

Explanation

To solve the limit \( \lim_{x \to 0^+} \frac{x}{a \left[ \frac{b}{x} \right]} \), where \( a \) and \( b \) are positive constants and \( [x] \) denotes the greatest integer less than or equal to \( x \), we can follow these steps:

### Step 1: Rewrite the greatest integer function
The greatest integer function can be expressed as:
\[ \left[ \frac{b}{x} \right] = \frac{b}{x} - \left\{ \frac{b}{x} \right\} \]
where \( \{y\} \) denotes the fractional part of \( y \). Thus, we can rewrite the limit as:
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