Given, `lim_(xto0) (sin3x)/(sin7x)`.

Options

A. `3/5`
B. `3/7`
C. `1/5`
D. `7/3`

Explanation

To solve the limit \( \lim_{x \to 0} \frac{\sin 3x}{\sin 7x} \), we can follow these steps:

### Step 1: Rewrite the Limit
We start with the limit expression:
\[ \lim_{x \to 0} \frac{\sin 3x}{\sin 7x} \]

### Step 2: Introduce Multiplication and Division by Constants
To facilitate the limit calculation, we can multiply and divide by \(3x\) and \(7x\):
\[ \lim_{x \to 0} \frac{\sin 3x}{\sin 7x} = \lim_{x \to 0} \frac{\sin 3x}{3x} \cdot \frac{3x}{7x} \cdot \frac{7x}{\sin 7x} \]
This can be rearranged as:
\[ = \lim_{x \to 0} \left( \frac{\sin 3x}{3x} \cdot \frac{3}{7} \cdot \frac{7x}{\sin 7x} \right) \]

### Step 3: Separate the Limits
We can separate the limit into three parts:
\[ = \frac{3}{7} \cdot \lim_{x \to 0} \frac{\sin 3x}{3x} \cdot \lim_{x \to 0} \frac{7x}{\sin 7x} \]

### Step 4: Apply the Standard Limit
We know from standard limit results that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
Thus, we can apply this result:
- For \( \lim_{x \to 0} \frac{\sin 3x}{3x} \), as \( x \to 0 \), \( 3x \to 0 \), so this limit approaches \(1\).
- For \( \lim_{x \to 0} \frac{7x}{\sin 7x} \), as \( x \to 0 \), \( 7x \to 0 \), so this limit also approaches \(1\).

### Step 5: Combine the Results
Putting it all together:
\[ = \frac{3}{7} \cdot 1 \cdot 1 = \frac{3}{7} \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\sin 3x}{\sin 7x} = \frac{3}{7} \]

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