`lim_(x to 0) (sin 3x)/(x)` is equal to

Options

A. 1
B. 3
C. 2
D. 0

Answer

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

### Step 1: Rewrite the limit
We start with the limit expression:
\[ \lim_{x \to 0} \frac{\sin(3x)}{x} \]

### Step 2: Introduce a substitution
To make use of the known limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), we can manipulate the expression. We can rewrite the limit as:
\[ \lim_{x \to 0} \frac{\sin(3x)}{3x} \cdot \frac{3x}{x} \]

### Step 3: Simplify the expression
This simplifies to:
\[ \lim_{x \to 0} \frac{\sin(3x)}{3x} \cdot 3 \]

### Step 4: Apply the known limit
Now, we can apply the known limit:
\[ \lim_{x \to 0} \frac{\sin(3x)}{3x} = 1 \]
Thus, we have:
\[ \lim_{x \to 0} \frac{\sin(3x)}{x} = 1 \cdot 3 = 3 \]

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

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