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Evaluate the limit lim_(x→ 0)((4^x-1)³)/(sin ((x²)/(4)) log(1+3x)).

1. (4)/(3)(ln 4)²
2. (4)/(3)(ln 4)³
3. (3)/(2)(ln 4)²
4. (3)/(2)(ln 4)³

Correct answer: (4)/(3)(ln 4)³

Solution

The limit evaluates to rac{4}{3}( ext{ln } 4)³ because the numerator approaches 0 at a cubic rate, while the denominator approaches 0 at a quadratic rate, leading to a limit that reflects the higher order of growth in the numerator.

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