Let `f(x) = x(-1)^([1//x]); x != 0` where [.] denotes greatest integer function, then `lim_(x rarr 0) f (x)` is :

Options

A. Does not exist
B. 2
C. 0
D. -1

Answer

To find the limit of the function \( f(x) = x(-1)^{[\frac{1}{x}]} \) as \( x \) approaches 0, we will analyze the behavior of the function from both the right-hand side (as \( x \to 0^+ \)) and the left-hand side (as \( x \to 0^- \)).

### Step 1: Right-hand limit as \( x \to 0^+ \)

We start by evaluating the right-hand limit:

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x(-1)^{[\frac{1}{x}]} \]

As \( x \) approaches 0 from the right, \( \frac{1}{x} \) approaches \( +\infty \). The greatest integer function \( [\frac{1}{x}] \) will take on increasingly large integer values. Specifically, it will be even or odd depending on the specific value of \( x \).

### Step 2: Behavior of \( (-1)^{[\frac{1}{x}]} \)

The term \( (-1)^{[\frac{1}{x}]} \) oscillates between 1 and -1 depending on whether \( [\frac{1}{x}] \) is even or odd. However, since \( x \) is approaching 0, the value of \( x \) itself will dominate the product:

\[ \lim_{x \to 0^+} x(-1)^{[\frac{1}{x}]} = \lim_{x \to 0^+} x \cdot \text{(oscillating term)} \]

### Step 3: Limit evaluation

Regardless of whether \( (-1)^{[\frac{1}{x}]} \) is 1 or -1, the product \( x \cdot (-1)^{[\frac{1}{x}]} \) will approach 0 because \( x \) approaches 0:

\[ \lim_{x \to 0^+} f(x) = 0 \]

### Step 4: Left-hand limit as \( x \to 0^- \)

Now, we evaluate the left-hand limit:

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x(-1)^{[\frac{1}{x}]} \]

As \( x \) approaches 0 from the left, \( \frac{1}{x} \) approaches \( -\infty \). The greatest integer function \( [\frac{1}{x}] \) will also take on increasingly large negative integer values, which means it will still oscillate between even and odd.

### Step 5: Behavior of \( (-1)^{[\frac{1}{x}]} \)

Similar to the right-hand limit, \( (-1)^{[\frac{1}{x}]} \) will oscillate between 1 and -1. Therefore, we have:

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x(-1)^{[\frac{1}{x}]} = 0 \]

### Step 6: Conclusion

Since both the right-hand limit and left-hand limit as \( x \) approaches 0 are equal:

\[ \lim_{x \to 0^+} f(x) = 0 \quad \text{and} \quad \lim_{x \to 0^-} f(x) = 0 \]

Thus, we conclude that:

\[ \lim_{x \to 0} f(x) = 0 \]

### Final Answer

The limit is \( \boxed{0} \).

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