`Lt_(xto0)(|sinx|)/(x)` is equal to (i) 1 (ii) `-1` (iii) does not exist (iv) none of these

Options

A. 1
B. `-1`
C. does not exist
D. none of these

Answer

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

### Step 1: Define the limit
We start with the limit expression:
\[ \lim_{x \to 0} \frac{|\sin x|}{x} \]

### Step 2: Consider the left-hand limit
We first evaluate the left-hand limit as \( x \) approaches 0 from the negative side (i.e., \( x \to 0^- \)):
\[ \lim_{x \to 0^-} \frac{|\sin x|}{x} \]
For \( x < 0 \), \( |\sin x| = -\sin x \). Thus, we can rewrite the limit:
\[ \lim_{x \to 0^-} \frac{|\sin x|}{x} = \lim_{x \to 0^-} \frac{-\sin x}{x} \]
Now, we know that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). Therefore:
\[ \lim_{x \to 0^-} \frac{-\sin x}{x} = -1 \]

### Step 3: Consider the right-hand limit
Next, we evaluate the right-hand limit as \( x \) approaches 0 from the positive side (i.e., \( x \to 0^+ \)):
\[ \lim_{x \to 0^+} \frac{|\sin x|}{x} \]
For \( x > 0 \), \( |\sin x| = \sin x \). Thus, we can rewrite the limit:
\[ \lim_{x \to 0^+} \frac{|\sin x|}{x} = \lim_{x \to 0^+} \frac{\sin x}{x} \]
Again, using the known limit:
\[ \lim_{x \to 0^+} \frac{\sin x}{x} = 1 \]

### Step 4: Compare the left-hand and right-hand limits
Now we compare the two limits we calculated:
- Left-hand limit: \( -1 \)
- Right-hand limit: \( 1 \)

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

### Final Answer
Thus, the correct option is:
(iii) does not exist.