If `f(x) = x(-1)^([1/x]), x le 0` where [x] denotes the greatest integer less than or equal to x, then the value of `lim_(x to 0) f(x)` is equal to:

Options

A. `-1`
B. 0
C. 2
D. none of these

Answer

To solve the limit \( \lim_{x \to 0} f(x) \) where \( f(x) = x(-1)^{[\frac{1}{x}]} \) and \( x \leq 0 \), we can follow these steps:

### Step 1: Understand the function
The function is defined as:
\[ f(x) = x(-1)^{[\frac{1}{x}]} \]
where \( [x] \) denotes the greatest integer less than or equal to \( x \).

### Step 2: Analyze \( \frac{1}{x} \) as \( x \to 0 \)
As \( x \) approaches 0 from the left (i.e., \( x \to 0^- \)), \( \frac{1}{x} \) approaches \( -\infty \). Therefore, we need to determine the behavior of \( [\frac{1}{x}] \).

### Step 3: Determine \( [\frac{1}{x}] \)
As \( x \) approaches 0 from the left, \( \frac{1}{x} \) becomes a large negative number. The greatest integer function \( [\frac{1}{x}] \) will be a large negative integer, say \( k \), where \( k \) is an integer and \( k \to -\infty \).

### Step 4: Analyze \( (-1)^{[\frac{1}{x}]} \)
The expression \( (-1)^{[\frac{1}{x}]} \) will alternate between 1 and -1 depending on whether \( [\frac{1}{x}] \) is even or odd. Since \( [\frac{1}{x}] \) can take any integer value as \( x \) approaches 0 from the left, it will oscillate between 1 and -1.

### Step 5: Evaluate the limit
Now we can express \( f(x) \):
\[ f(x) = x(-1)^{[\frac{1}{x}]} \]
As \( x \to 0^- \), \( x \) approaches 0 and \( (-1)^{[\frac{1}{x}]} \) oscillates between 1 and -1. Therefore, regardless of whether \( (-1)^{[\frac{1}{x}]} \) is 1 or -1, the term \( x \) will dominate the behavior of \( f(x) \).

Thus, we have:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x(-1)^{[\frac{1}{x}]} = 0 \]

### Final Answer
Therefore, the value of \( \lim_{x \to 0} f(x) \) is:
\[ \boxed{0} \]