`lim_(xto0) [(1-e^(x))(sinx)/(|x|)]` is (where `[.]` represents the greatest integer function )

Options

A. 1
B. 2
C. 3
D. none of these

Explanation

To solve the limit \( \lim_{x \to 0} \frac{(1 - e^x) \sin x}{|x|} \) and find the greatest integer function of the result, we will evaluate the left-hand limit and the right-hand limit separately.

### Step 1: Define the limit
We need to evaluate the limit as \( x \) approaches 0. We will consider two cases: when \( x \) approaches 0 from the left (negative) and from the right (positive).

### Step 2: Evaluate the left-hand limit
For \( x < 0 \):
- The modulus function \( |x| = -x \).
- Thus, we rewrite the limit:
\[ \lim_{x \to 0^-} \frac{(1 - e^x) \sin x}{|x|} = \lim_{x \to 0^-} \frac{(1 - e^x) \sin x}{-x} \]

### Step 3: Simplify the left-hand limit
Now, we can evaluate the limit:
\[ \lim_{x \to 0^-} \frac{(1 - e^x) \sin x}{-x} = -\lim_{x \to 0^-} \frac{(1 - e^x) \sin x}{x} \]
Using the fact that \( \sin x \approx x \) as \( x \to 0 \), we can substitute:
\[ \lim_{x \to 0^-} \frac{(1 - e^x) \sin x}{x} = \lim_{x \to 0^-} (1 - e^x) \cdot \frac{\sin x}{x} \]
As \( x \to 0 \), \( \frac{\sin x}{x} \to 1 \).

### Step 4: Evaluate \( 1 - e^x \)
As \( x \to 0^- \), \( e^x \to 1 \), so:
\[ 1 - e^x \to 1 - 1 = 0 \]
Thus, we have:
\[ \lim_{x \to 0^-} (1 - e^x) \cdot 1 = 0 \]

### Step 5: Evaluate the right-hand limit
For \( x > 0 \):
- The modulus function \( |x| = x \).
- Thus, we rewrite the limit:
\[ \lim_{x \to 0^+} \frac{(1 - e^x) \sin x}{|x|} = \lim_{x \to 0^+} \frac{(1 - e^x) \sin x}{x} \]

### Step 6: Simplify the right-hand limit
Using the same reasoning as before:
\[ \lim_{x \to 0^+} \frac{(1 - e^x) \sin x}{x} = (1 - e^0) \cdot 1 = 0 \]

### Step 7: Combine the results
Both the left-hand limit and the right-hand limit are equal to 0:
\[ \lim_{x \to 0^-} \frac{(1 - e^x) \sin x}{|x|} = 0 \]
\[ \lim_{x \to 0^+} \frac{(1 - e^x) \sin x}{|x|} = 0 \]

### Step 8: Final limit
Thus, we conclude:
\[ \lim_{x \to 0} \frac{(1 - e^x) \sin x}{|x|} = 0 \]

### Step 9: Apply the greatest integer function
Now we apply the greatest integer function:
\[ \lfloor 0 \rfloor = 0 \]

### Final Answer
The final answer is:
\[ \boxed{0} \]