`lim_(xrarr(-1)/(3)) (1)/(x)[(-1)/(x)]=` (where [.] denotes the greatest integer function)

Options

A. `-9`
B. `-12`
C. `-6`
D. 0

Explanation

To solve the limit problem \( \lim_{x \to -\frac{1}{3}} \frac{1}{x} \left[ -\frac{1}{x} \right] \), where \([.]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Identify the limit point
We need to evaluate the limit as \( x \) approaches \( -\frac{1}{3} \). Since \( x \) is approaching \( -\frac{1}{3} \), we will consider values of \( x \) that are less than \( -\frac{1}{3} \).

### Step 2: Calculate \( \frac{1}{x} \)
As \( x \) approaches \( -\frac{1}{3} \) from the left (i.e., \( x < -\frac{1}{3} \)), we can compute:
\[ \frac{1}{x} \to \frac{1}{-\frac{1}{3}} = -3 \]
Thus, \( \frac{1}{x} \) will be greater than \( -3 \) for \( x < -\frac{1}{3} \).

### Step 3: Calculate \( -\frac{1}{x} \)
Next, we compute:
\[ -\frac{1}{x} \to -\left(-3\right) = 3 \]
This means that as \( x \) approaches \( -\frac{1}{3} \), \( -\frac{1}{x} \) approaches \( 3 \).

### Step 4: Apply the greatest integer function
Now we need to evaluate \( \left[-\frac{1}{x}\right] \). Since \( -\frac{1}{x} \) approaches \( 3 \) from the left (as \( x \) approaches \( -\frac{1}{3} \)), the greatest integer function will yield:
\[ \left[-\frac{1}{x}\right] = 2 \]
because the greatest integer less than \( 3 \) is \( 2 \).

### Step 5: Substitute back into the limit
Now we substitute back into our limit:
\[ \lim_{x \to -\frac{1}{3}} \frac{1}{x} \left[-\frac{1}{x}\right] = \lim_{x \to -\frac{1}{3}} \frac{1}{x} \cdot 2 \]
As \( x \) approaches \( -\frac{1}{3} \):
\[ \frac{1}{x} \to -3 \]
Thus, we have:
\[ \lim_{x \to -\frac{1}{3}} \frac{1}{x} \cdot 2 = -3 \cdot 2 = -6 \]

### Final Answer
Therefore, the limit is:
\[ \boxed{-6} \]