`lim(x->a^-) {(|x|^3)/a-[x/a]^3} ,(a > 0)`, where `[x]` denotes the greatest integer less than or equal to `x` is equal to:

Options

A. `a^2-3`
B. `a^2-1`
C. `a^2`
D. none of these

Explanation

To solve the limit \( \lim_{x \to a^-} \left( \frac{|x|^3}{a} - \left[\frac{x}{a}\right]^3 \right) \) where \( a > 0 \), we will break it down step by step.

### Step 1: Analyze the limit as \( x \) approaches \( a \) from the left
As \( x \to a^- \), \( x \) is approaching \( a \) but is always less than \( a \). Therefore, we can express \( x \) as \( x = a - \epsilon \) where \( \epsilon \) is a small positive number.

### Step 2: Evaluate \( |x|^3 \)
Since \( a > 0 \) and \( x \) is approaching \( a \) from the left, we have:
\[ |x| = x \quad \text{(because \( x \) is positive)} \]
Thus,
\[ |x|^3 = x^3 = (a - \epsilon)^3 \]

### Step 3: Expand \( (a - \epsilon)^3 \)
Using the binomial expansion:
\[ (a - \epsilon)^3 = a^3 - 3a^2\epsilon + 3a\epsilon^2 - \epsilon^3 \]

### Step 4: Evaluate \( \left[\frac{x}{a}\right] \)
Now, we need to evaluate \( \left[\frac{x}{a}\right] \):
\[ \frac{x}{a} = \frac{a - \epsilon}{a} = 1 - \frac{\epsilon}{a} \]
Since \( \epsilon \) is a small positive number, \( \frac{\epsilon}{a} \) is also small and positive. Therefore:
\[ \left[\frac{x}{a}\right] = 0 \quad \text{(as it is the greatest integer less than or equal to \( 1 - \frac{\epsilon}{a} \))} \]

### Step 5: Substitute into the limit expression
Now substituting back into the limit:
\[ \lim_{x \to a^-} \left( \frac{(a - \epsilon)^3}{a} - 0^3 \right) = \lim_{x \to a^-} \frac{(a - \epsilon)^3}{a} \]
This simplifies to:
\[ \lim_{x \to a^-} \frac{a^3 - 3a^2\epsilon + 3a\epsilon^2 - \epsilon^3}{a} \]

### Step 6: Simplify the expression
Breaking it down:
\[ \frac{(a - \epsilon)^3}{a} = \frac{a^3}{a} - \frac{3a^2\epsilon}{a} + \frac{3a\epsilon^2}{a} - \frac{\epsilon^3}{a} \]
This simplifies to:
\[ a^2 - 3\epsilon + 3\frac{\epsilon^2}{a} - \frac{\epsilon^3}{a} \]

### Step 7: Take the limit as \( \epsilon \to 0 \)
As \( \epsilon \to 0 \):
\[ \lim_{\epsilon \to 0} \left( a^2 - 3\epsilon + 3\frac{\epsilon^2}{a} - \frac{\epsilon^3}{a} \right) = a^2 \]

### Conclusion
Thus, the limit is:
\[ \lim_{x \to a^-} \left( \frac{|x|^3}{a} - \left[\frac{x}{a}\right]^3 \right) = a^2 \]

### Final Answer
The limit is \( a^2 \).

---