Evaluate `lim_(x to 0) ("sin"5x)/("sin"7x)`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\sin(5x)}{\sin(7x)} \), we can follow these steps:

### Step 1: Rewrite the Limit
We start with the limit:
\[ \lim_{x \to 0} \frac{\sin(5x)}{\sin(7x)} \]

### Step 2: Multiply and Divide by \(5x\) and \(7x\)
To use the standard limit \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \), we can multiply and divide the expression by \(5x\) and \(7x\):
\[ \lim_{x \to 0} \frac{\sin(5x)}{\sin(7x)} \cdot \frac{5x}{5x} \cdot \frac{7x}{7x} = \lim_{x \to 0} \frac{\sin(5x)}{5x} \cdot \frac{5x}{7x} \cdot \frac{7x}{\sin(7x)} \]

### Step 3: Simplify the Expression
Now we can rewrite the limit:
\[ \lim_{x \to 0} \left( \frac{\sin(5x)}{5x} \cdot \frac{5}{7} \cdot \frac{7x}{\sin(7x)} \right) \]

### Step 4: Evaluate the Limits
Using the known limit \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \):
- As \(x \to 0\), \( \frac{\sin(5x)}{5x} \to 1 \)
- As \(x \to 0\), \( \frac{7x}{\sin(7x)} \to 1 \)

Thus, we have:
\[ \lim_{x \to 0} \frac{\sin(5x)}{5x} = 1 \quad \text{and} \quad \lim_{x \to 0} \frac{7x}{\sin(7x)} = 1 \]

### Step 5: Combine the Results
Putting it all together:
\[ \lim_{x \to 0} \frac{\sin(5x)}{\sin(7x)} = 1 \cdot \frac{5}{7} \cdot 1 = \frac{5}{7} \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\sin(5x)}{\sin(7x)} = \frac{5}{7} \]

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