Evaluate the following :
`lim_(x to 0)(sin5x)/(sin3x)`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\sin(5x)}{\sin(3x)} \), we can use the standard limit property that states:

\[ \lim_{x \to 0} \frac{\sin(ax)}{ax} = 1 \]

for any constant \( a \).

### Step-by-Step Solution:

1. **Rewrite the limit**:
We can express the limit in a form that allows us to apply the standard limit property. We can multiply and divide by \( 5x \) and \( 3x \):

\[ \lim_{x \to 0} \frac{\sin(5x)}{\sin(3x)} = \lim_{x \to 0} \frac{\sin(5x)}{5x} \cdot \frac{5x}{3x} \cdot \frac{3x}{\sin(3x)} \]

2. **Separate the limit**:
This can be separated into two limits:

\[ = \lim_{x \to 0} \frac{\sin(5x)}{5x} \cdot \frac{5}{3} \cdot \lim_{x \to 0} \frac{3x}{\sin(3x)} \]

3. **Evaluate the first limit**:
From the standard limit property, we know:

\[ \lim_{x \to 0} \frac{\sin(5x)}{5x} = 1 \]

4. **Evaluate the second limit**:
Similarly, we also have:

\[ \lim_{x \to 0} \frac{3x}{\sin(3x)} = 1 \]

5. **Combine the results**:
Now we can combine the results from the limits:

\[ = 1 \cdot \frac{5}{3} \cdot 1 = \frac{5}{3} \]

### Final Answer:
Thus, the limit evaluates to:

\[ \lim_{x \to 0} \frac{\sin(5x)}{\sin(3x)} = \frac{5}{3} \]