`Lt_(xto0)(x)/(sin3x)` is equal to
(i) 3
(ii) `1/3`
(iii) 0
(iv) 1

Options

A. 3
B. `1/3`
C. 0
D. 1

Answer

To solve the limit \( \lim_{x \to 0} \frac{x}{\sin(3x)} \), we can follow these steps:

### Step 1: Rewrite the limit
We can rewrite the limit as:
\[ \lim_{x \to 0} \frac{x}{\sin(3x)} = \lim_{x \to 0} \frac{1}{\frac{\sin(3x)}{x}} \]

### Step 2: Adjust the expression
Next, we can manipulate the expression inside the limit:
\[ \lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \frac{\sin(3x)}{3x} \cdot 3 \]
This allows us to separate the constant factor of 3.

### Step 3: Apply the standard limit
Using the standard limit result \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \), we can substitute \( u = 3x \):
\[ \lim_{x \to 0} \frac{\sin(3x)}{3x} = 1 \]

### Step 4: Combine the results
Now we can combine everything:
\[ \lim_{x \to 0} \frac{x}{\sin(3x)} = \lim_{x \to 0} \frac{1}{3 \cdot \frac{\sin(3x)}{3x}} = \frac{1}{3 \cdot 1} = \frac{1}{3} \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{x}{\sin(3x)} = \frac{1}{3} \]