`lim_(xrarr0)(|sinx|)/(x)` is equal to

Options

A. 1
B. `-1`
C. Does not exist
D. None of these

Explanation

To evaluate the limit \( \lim_{x \to 0} \frac{|\sin x|}{x} \), we can break it down into steps.

### Step 1: Understand the absolute value function
The absolute value function \( |\sin x| \) behaves differently depending on the sign of \( \sin x \):
- When \( x \geq 0 \), \( |\sin x| = \sin x \).
- When \( x < 0 \), \( |\sin x| = -\sin x \).

### Step 2: Define the limit in terms of left-hand and right-hand limits
To find the limit as \( x \) approaches 0, we can consider the left-hand limit (as \( x \) approaches 0 from the negative side) and the right-hand limit (as \( x \) approaches 0 from the positive side).

### Step 3: Calculate the right-hand limit
For \( x \geq 0 \):
\[ \lim_{x \to 0^+} \frac{|\sin x|}{x} = \lim_{x \to 0^+} \frac{\sin x}{x} \]
Using the standard limit result \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \):
\[ \lim_{x \to 0^+} \frac{|\sin x|}{x} = 1 \]

### Step 4: Calculate the left-hand limit
For \( x < 0 \):
\[ \lim_{x \to 0^-} \frac{|\sin x|}{x} = \lim_{x \to 0^-} \frac{-\sin x}{x} \]
Again, using the standard limit result:
\[ \lim_{x \to 0^-} \frac{-\sin x}{x} = -1 \]

### Step 5: Compare the left-hand and right-hand limits
Now we have:
- Right-hand limit: \( 1 \)
- Left-hand limit: \( -1 \)

Since the left-hand limit does not equal the right-hand limit, we conclude that:
\[ \lim_{x \to 0} \frac{|\sin x|}{x} \text{ does not exist.} \]

### Final Answer:
The limit \( \lim_{x \to 0} \frac{|\sin x|}{x} \) does not exist.

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