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

Answer

To evaluate the limit

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

we can follow these steps:

### Step 1: Rewrite the sine function
We know from the sine addition formula that

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

### Step 2: Substitute the sine function
Substituting this into our limit gives us:

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

### Step 3: Simplify the expression
This simplifies to:

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

### Step 4: Factor out constants
We can factor out the constant:

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

### Step 5: Use the limit property
Using the limit property \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\), we can rewrite the limit as:

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

### Step 6: Evaluate the limit
Thus, we have:

\[ = 4 \cdot 1^3 = 4. \]

### Final Answer
Therefore, the limit is

\[ \boxed{4}. \]

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