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The value of lim(x->0) [1 - cos(sin 3x)] / [ln(1 + 4x) * (e^(2x) - 1)] is

1. 9/8
2. 4/9
3. 8/9
4. 9/16

Correct answer: 9/16

Solution

Substituting leading-order approximations: numerator ~ (3x)²/2 = 9x²/2; denominator ~ (4x)(2x) = 8x². The limit equals (9/2)/8 = 9/16.

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