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

Answer

To evaluate the limit

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

we can follow these steps:

### Step 1: Identify the form of the limit
First, we substitute \(x = 0\) into the expression:

\[ \frac{\sin(3 \cdot 0) - \sin(0)}{\sin(0)} = \frac{\sin(0) - \sin(0)}{\sin(0)} = \frac{0 - 0}{0} = \frac{0}{0}. \]

This is an indeterminate form \(0/0\).

**Hint:** When you encounter a \(0/0\) form, consider using algebraic manipulation or L'Hôpital's Rule.

### Step 2: Use the sine addition formula
Instead of applying L'Hôpital's Rule, we can use the sine subtraction formula. We know that:

\[ \sin(3x) = 3\sin(x) - 4\sin^3(x). \]

Now we can rewrite the limit:

\[ \lim_{x \to 0} \frac{(3\sin(x) - 4\sin^3(x)) - \sin(x)}{\sin(x)}. \]

**Hint:** Look for ways to simplify the expression by combining like terms.

### Step 3: Simplify the expression
Now, simplify the numerator:

\[ 3\sin(x) - 4\sin^3(x) - \sin(x) = (3\sin(x) - \sin(x)) - 4\sin^3(x) = 2\sin(x) - 4\sin^3(x). \]

Thus, the limit becomes:

\[ \lim_{x \to 0} \frac{2\sin(x) - 4\sin^3(x)}{\sin(x)}. \]

**Hint:** Factor out common terms in the numerator.

### Step 4: Factor out \(\sin(x)\)
We can factor \(\sin(x)\) out of the numerator:

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

Now, we can cancel \(\sin(x)\) (since \(\sin(x) \neq 0\) as \(x\) approaches 0):

\[ = \lim_{x \to 0} (2 - 4\sin^2(x)). \]

**Hint:** Now that you have simplified the limit, you can directly substitute \(x = 0\).

### Step 5: Substitute \(x = 0\)
Now, substitute \(x = 0\):

\[ = 2 - 4\sin^2(0) = 2 - 4 \cdot 0 = 2. \]

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

\[ \boxed{2}. \]