`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 analyze the behavior of the expression as \( x \) approaches 0 from both the right and the left.

### Step 1: Evaluate the Right-Hand Limit (RHL)
We start by considering the limit as \( x \) approaches 0 from the right (denoted as \( 0^+ \)):
\[ \lim_{x \to 0^+} \frac{(1 - e^x)(\sin x)}{|x|} = \lim_{x \to 0^+} \frac{(1 - e^x)(\sin x)}{x} \]
Here, since \( x \) is positive, \( |x| = x \).

### Step 2: Analyze \( 1 - e^x \)
As \( x \to 0^+ \), we can use the Taylor expansion for \( e^x \):
\[ e^x \approx 1 + x \quad \text{(for small \( x \))} \]
Thus,
\[ 1 - e^x \approx 1 - (1 + x) = -x \]

### Step 3: Analyze \( \sin x \)
Using the Taylor expansion for \( \sin x \):
\[ \sin x \approx x \quad \text{(for small \( x \))} \]

### Step 4: Substitute Back into the Limit
Substituting these approximations into the limit:
\[ \lim_{x \to 0^+} \frac{(-x)(x)}{x} = \lim_{x \to 0^+} -x = 0 \]

### Step 5: Evaluate the Left-Hand Limit (LHL)
Now, we consider the limit as \( x \) approaches 0 from the left (denoted as \( 0^- \)):
\[ \lim_{x \to 0^-} \frac{(1 - e^x)(\sin x)}{|x|} = \lim_{x \to 0^-} \frac{(1 - e^x)(\sin x)}{-x} \]
Here, since \( x \) is negative, \( |x| = -x \).

### Step 6: Analyze \( 1 - e^x \) Again
As \( x \to 0^- \), we still have:
\[ 1 - e^x \approx -x \]

### Step 7: Analyze \( \sin x \)
For \( \sin x \), it still holds that:
\[ \sin x \approx x \]

### Step 8: Substitute Back into the Limit
Substituting these approximations into the limit:
\[ \lim_{x \to 0^-} \frac{(-x)(x)}{-x} = \lim_{x \to 0^-} x = 0 \]

### Step 9: Conclusion
Since both the right-hand limit and the left-hand limit approach 0, we conclude:
\[ \lim_{x \to 0} \frac{(1 - e^x)(\sin x)}{|x|} = 0 \]

### Step 10: Apply the Greatest Integer Function
The greatest integer function \( [0] \) is:
\[ [0] = 0 \]

Thus, the final answer is:
\[ \boxed{0} \]