Evaluate the following limits :
`Lim_(x to 0 ) ( sin 5x - sin 3x )/(sin x )`

Answer

To evaluate the limit

\[ \lim_{x \to 0} \frac{\sin 5x - \sin 3x}{\sin x}, \]

we can use the sine subtraction formula.

### Step 1: Apply the sine subtraction formula

The sine subtraction formula states that

\[ \sin a - \sin b = 2 \cos\left(\frac{a + b}{2}\right) \sin\left(\frac{a - b}{2}\right). \]

In our case, let \( a = 5x \) and \( b = 3x \). Thus, we have:

\[ \sin 5x - \sin 3x = 2 \cos\left(\frac{5x + 3x}{2}\right) \sin\left(\frac{5x - 3x}{2}\right) = 2 \cos(4x) \sin(x). \]

### Step 2: Substitute into the limit

Now we can substitute this back into our limit:

\[ \lim_{x \to 0} \frac{2 \cos(4x) \sin(x)}{\sin x}. \]

### Step 3: Simplify the expression

The \(\sin x\) in the numerator and denominator cancels out:

\[ \lim_{x \to 0} 2 \cos(4x). \]

### Step 4: Evaluate the limit

Now we can evaluate the limit as \( x \) approaches 0:

\[ 2 \cos(4 \cdot 0) = 2 \cos(0) = 2 \cdot 1 = 2. \]

Thus, the final answer is

\[ \boxed{2}. \]

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