What is the value of $$\lim_{x \to 0} \frac{\sin 3x \cos 3x}{12x}?$$ - HSC - SSCE Mathematics Extension 1 - Question 5 - 2018 - Paper 1

To find the limit as $x$ approaches 0, we can simplify the expression:

1. Identify the Limit: Start with the expression: $\lim_{x \to 0} \frac{\sin 3x \cos 3x}{12x}$

2. Using Trigonometric Limits: Recall that as $x \to 0$,
   $\sin kx \approx kx$ for small values of $x$. Therefore, $\sin 3x \approx 3x$ and since $\cos 3x \to 1$ as $x \to 0$, we replace to get: $\lim_{x \to 0} \frac{3x \cdot 1}{12x}$

3. Simplifying the Expression: Substitute: $\lim_{x \to 0} \frac{3x}{12x} = \lim_{x \to 0} \frac{3}{12} = \frac{1}{4}$

Thus, the value of the limit is ( \frac{1}{4} ).