If $\lim_{x \to 0} \frac{a \sin x - \sin 2x}{\tan^3 x} = 1$, find $a$.
Step-by-Step Solution
Key Concept: Numerator $\approx ax - 2x \cos x \approx (a - 2)x + x^3 + \dots$. For a finite non-zero limit at $x^3$, need $a - 2 = 0 \Rightarrow a = 2$. Then: $(2 \sin x - \sin 2x)/\tan^3 x \approx 2 \sin x(1 - \cos x)/x^3 \approx 2x \cdot x^2/2/x^3 = 1$.
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Correct Answer: 2