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 use the multiple angle formula for sine. The formula states that

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

### Step 1: Substitute the multiple angle formula
We can substitute \(\sin 3x\) in the limit expression:

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

### Step 2: Simplify the expression
Now simplify the expression inside the limit:

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

Thus, we have:

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

### Step 3: Rewrite the limit
We can rewrite the limit as:

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

### Step 4: Apply the standard limit
We know from standard calculus that:

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1. \]

Thus,

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

### Step 5: Final calculation
Now we can substitute this back into our limit:

\[ 4 \cdot 1 = 4. \]

### Conclusion
Therefore, the limit evaluates to:

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

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