The value of the `lim_(x->0)x/a[b/x](a!=0)(where [*]` denotes the greatest integer function) is

Options

A. a
B. b
C. `b/a`
D. `1-b/a`

Answer

To solve the limit \( \lim_{x \to 0} \frac{x}{a \left[ \frac{b}{x} \right]} \) where \( a \neq 0 \) and \( [\cdot] \) denotes the greatest integer function, we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \lim_{x \to 0} \frac{x}{a \left[ \frac{b}{x} \right]} \]
We know that the greatest integer function \( \left[ \frac{b}{x} \right] \) can be rewritten as:
\[ \left[ \frac{b}{x} \right] = \frac{b}{x} - \left\{ \frac{b}{x} \right\} \]
where \( \{ \cdot \} \) denotes the fractional part. Thus, we can express our limit as:
\[ \lim_{x \to 0} \frac{x}{a \left( \frac{b}{x} - \left\{ \frac{b}{x} \right\} \right)} \]

### Step 2: Simplify the Expression
Substituting this back into our limit gives:
\[ \lim_{x \to 0} \frac{x}{a \left( \frac{b}{x} - \left\{ \frac{b}{x} \right\} \right)} = \lim_{x \to 0} \frac{x}{\frac{ab}{x} - a \left\{ \frac{b}{x} \right\}} \]
Now, multiplying the numerator and denominator by \( x \) yields:
\[ \lim_{x \to 0} \frac{x^2}{ab - a x \left\{ \frac{b}{x} \right\}} \]

### Step 3: Analyze the Limit
As \( x \to 0 \), \( \frac{b}{x} \to \infty \), which means \( \left\{ \frac{b}{x} \right\} \) oscillates between 0 and 1. Therefore, \( a x \left\{ \frac{b}{x} \right\} \) will approach 0 since \( x \) approaches 0. Thus, the limit simplifies to:
\[ \lim_{x \to 0} \frac{x^2}{ab - 0} = \lim_{x \to 0} \frac{x^2}{ab} \]

### Step 4: Evaluate the Final Limit
Since \( ab \) is a constant (as \( a \neq 0 \)), we have:
\[ \lim_{x \to 0} \frac{x^2}{ab} = 0 \]

### Conclusion
Thus, the value of the limit is:
\[ \boxed{0} \]