`lim_(xto0)(sin|x|)/(x)` equal to ?

Options

A. 1
B. `-1`
C. `oo`
D. Limit D.N.E.

Answer

To solve the limit \( \lim_{x \to 0} \frac{\sin |x|}{x} \), we need to analyze the behavior of the function as \( x \) approaches 0 from both the left and the right.

### Step-by-Step Solution:

1. **Understanding the Absolute Value**:
The expression \( |x| \) means that:
- If \( x \) is positive (approaching 0 from the right), then \( |x| = x \).
- If \( x \) is negative (approaching 0 from the left), then \( |x| = -x \).

2. **Split the Limit into Two Cases**:
We will evaluate the limit from the left (denoted as \( 0^- \)) and from the right (denoted as \( 0^+ \)):
- **Left-hand limit**: \( \lim_{x \to 0^-} \frac{\sin |x|}{x} = \lim_{x \to 0^-} \frac{\sin (-x)}{x} = \lim_{x \to 0^-} \frac{-\sin x}{x} \)
- **Right-hand limit**: \( \lim_{x \to 0^+} \frac{\sin |x|}{x} = \lim_{x \to 0^+} \frac{\sin x}{x} \)

3. **Evaluate the Left-hand Limit**:
- We know that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).
- Therefore, \( \lim_{x \to 0^-} \frac{-\sin x}{x} = -1 \).

4. **Evaluate the Right-hand Limit**:
- Similarly, \( \lim_{x \to 0^+} \frac{\sin x}{x} = 1 \).

5. **Compare the Limits**:
- Left-hand limit: \( \lim_{x \to 0^-} \frac{\sin |x|}{x} = -1 \)
- Right-hand limit: \( \lim_{x \to 0^+} \frac{\sin |x|}{x} = 1 \)

6. **Conclusion**:
Since the left-hand limit and right-hand limit are not equal:
\[ \lim_{x \to 0} \frac{\sin |x|}{x} \text{ does not exist.} \]

### Final Answer:
The limit does not exist.