What is the value of $$rac{ ext{sin}(3x) ext{cos}(3x)}{12x}?$$ A - HSC - SSCE Mathematics Extension 1 - Question 5 - 2018 - Paper 1

To find the limit as $x$ approaches $0$, we can use the fact that:

1. The limit of rac{ ext{sin}(u)}{u} as $u$ approaches $0$ is $1$. In this case, let $u = 3x$ so that as $x$ approaches $0$, $u$ also approaches $0$. Thus, we can rewrite: rac{ ext{sin}(3x)}{3x} \to 1

2. The expression can be rewritten in terms of $u$: rac{ ext{sin}(3x)\cos(3x)}{12x} = rac{\frac{ ext{sin}(3x)}{3x} \cdot \cos(3x)}{12/3} = \frac{3}{12}\cdot rac{ ext{sin}(3x)}{3x} \cdot ext{cos}(3x)

3. Next, we evaluate: $ext{cos}(3x) \to ext{cos}(0) = 1$

4. Therefore, rac{3}{12} \cdot 1 = \frac{1}{4}

Thus, the limit is rac{1}{4}.