The value of $\lim_{x \to 0} \frac{\sin(\tan x)}{x(2 + \cos(3x))}$ is equal to
Step-by-Step Solution
Key Concept: Apply standard limits for sine and tangent functions by factoring out appropriate terms
We need to find $\lim_{x \to 0} \frac{\sin 6x}{x(\tan 6x)}$. Using L'Hospital's rule and the standard limits $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \frac{\tan x}{x} = 1$, we can rewrite this as $\lim_{x \to 0} \frac{\sin 6x}{6x} \cdot \frac{1}{\tan 6x/6x} = 1 \cdot \frac{1}{1} = 1$.
Correct Answer: 1