The left hand limit of `f(x) = {|x|^(3)/a -[x/a]^(3)}, (a gt 0)`
where [x] denotes the greatest integer less than or equal to x is

Options

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

Answer

To find the left-hand limit of the function \( f(x) = \frac{|x|^3}{a} - \left[\frac{x}{a}\right]^3 \) as \( x \) approaches \( a \) from the left (denoted as \( x \to a^- \)), we will follow these steps:

### Step 1: Understand the Function
The function is given as:
\[ f(x) = \frac{|x|^3}{a} - \left[\frac{x}{a}\right]^3 \]
Since we are considering the limit as \( x \) approaches \( a \) from the left, we note that \( x \) will be slightly less than \( a \) (i.e., \( x \to a^- \)).

### Step 2: Evaluate \( |x| \) and \( \left[\frac{x}{a}\right] \)
For \( x < a \) (as \( x \to a^- \)):
- \( |x| = x \) because \( x \) is positive.
- \( \left[\frac{x}{a}\right] \) is the greatest integer less than or equal to \( \frac{x}{a} \). Since \( x < a \), \( \frac{x}{a} < 1 \), thus \( \left[\frac{x}{a}\right] = 0 \).

### Step 3: Substitute into the Function
Substituting these values into the function:
\[ f(x) = \frac{x^3}{a} - 0^3 = \frac{x^3}{a} \]

### Step 4: Calculate the Limit
Now we need to find the limit as \( x \) approaches \( a \) from the left:
\[ \lim_{x \to a^-} f(x) = \lim_{x \to a^-} \frac{x^3}{a} \]
Substituting \( x = a \):
\[ = \frac{a^3}{a} = a^2 \]

### Conclusion
Thus, the left-hand limit of \( f(x) \) as \( x \) approaches \( a \) from the left is:
\[ \boxed{a^2} \]