Evaluate the following limits :
`lim_(theta to 0)(sin5theta-sin3theta)/(sintheta)`.

Answer

To evaluate the limit

\[ \lim_{\theta \to 0} \frac{\sin(5\theta) - \sin(3\theta)}{\sin(\theta)}, \]

we start by substituting \(\theta = 0\):

1. **Substitution**:
\[ \frac{\sin(5 \cdot 0) - \sin(3 \cdot 0)}{\sin(0)} = \frac{0 - 0}{0} = \frac{0}{0}. \]
This is an indeterminate form \(0/0\), so we need to manipulate the expression.

2. **Rewrite the limit**:
We can rewrite the limit as:
\[ \lim_{\theta \to 0} \left( \frac{\sin(5\theta)}{\sin(\theta)} - \frac{\sin(3\theta)}{\sin(\theta)} \right). \]

3. **Separate the limits**:
This can be separated into two limits:
\[ \lim_{\theta \to 0} \frac{\sin(5\theta)}{\sin(\theta)} - \lim_{\theta \to 0} \frac{\sin(3\theta)}{\sin(\theta)}. \]

4. **Use the limit property**:
We know that:
\[ \lim_{x \to 0} \frac{\sin(kx)}{kx} = 1 \quad \text{for any constant } k. \]
Therefore, we can express the limits as:
\[ \lim_{\theta \to 0} \frac{\sin(5\theta)}{\sin(\theta)} = \lim_{\theta \to 0} \frac{\sin(5\theta)}{5\theta} \cdot \frac{5\theta}{\sin(\theta)} = 5 \cdot 1 = 5, \]
and
\[ \lim_{\theta \to 0} \frac{\sin(3\theta)}{\sin(\theta)} = \lim_{\theta \to 0} \frac{\sin(3\theta)}{3\theta} \cdot \frac{3\theta}{\sin(\theta)} = 3 \cdot 1 = 3. \]

5. **Combine the results**:
Now substituting back into our separated limits:
\[ 5 - 3 = 2. \]

Thus, the limit evaluates to:

\[ \boxed{2}. \]