If `f(x)=x((e^(|x|+[x])-2)/(|x|+[x]))` then (where [.] represent the greatest integer function)

Options

A. `underset(xrarr0^(+))(lim)f(x)=-1`
B. `underset(xrarr0^(-))(lim)f(x)=0`
C. `underset(xrarr0)(lim)f(0)=-1`
D. `underset(xrarr0)(lim)f(x)=0`

Explanation

To solve the problem, we need to evaluate the limits of the function \( f(x) = x \frac{e^{|x| + [x]} - 2}{|x| + [x]} \) as \( x \) approaches 0 from the positive and negative sides. Here, \([x]\) denotes the greatest integer function.

### Step 1: Evaluate the limit as \( x \to 0^+ \)

1. **Substitute into the function**:
\[ f(x) = x \frac{e^{|x| + [x]} - 2}{|x| + [x]} \]
For \( x \to 0^+ \):
- \( |x| = x \)
- \( [x] = 0 \) (since \( x \) is positive and less than 1)

Thus, the function simplifies to:
\[ f(x) = x \frac{e^{x + 0} - 2}{x + 0} = x \frac{e^x - 2}{x} \]

2. **Simplify the expression**:
\[ f(x) = e^x - 2 \]

3. **Take the limit**:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (e^x - 2) = e^0 - 2 = 1 - 2 = -1 \]

### Step 2: Evaluate the limit as \( x \to 0^- \)

1. **Substitute into the function**:
For \( x \to 0^- \):
- \( |x| = -x \)
- \( [x] = -1 \) (since \( x \) is negative and greater than -1)

Thus, the function simplifies to:
\[ f(x) = x \frac{e^{-x - 1} - 2}{-x - 1} \]

2. **Simplify the expression**:
\[ f(x) = x \frac{e^{-x - 1} - 2}{-x - 1} \]

3. **Take the limit**:
As \( x \to 0^- \):
\[ f(x) = x \frac{e^{-1} e^x - 2}{-1} = -x (e^{-1} e^x - 2) \]
Evaluating the limit:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -x (e^{-1} e^x - 2) = 0 \cdot (e^{-1} - 2) = 0 \]

### Final Results

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

### Conclusion

Based on the evaluations:
- Option 1: \( \lim_{x \to 0^+} f(x) = -1 \) is correct.
- Option 2: \( \lim_{x \to 0^-} f(x) = 0 \) is correct.