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Let f(x) = arctan(2 * ln(cos(arcsin(2x - x²)))). Find the value of lim(x->0) (e^(f(x)) - 1) / (cos x - 1).

1. 0
2. 4
3. 8
4. -8

Correct answer: 8

Solution

At x->0: 2x - x² ~ 2x. arcsin(2x) ~ 2x. cos(arcsin(2x)) = sqrt(1-4x²) ~ 1 - 2x². ln(1-2x²) ~ -2x². 2*ln(...) ~ -4x². f(x) = arctan(-4x²) ~ -4x². Numerator: e^(f(x))-1 ~ f(x) ~ -4x². Denominator: cos x - 1 ~ -x²/2. Limit = (-4x²)/(-x²/2) = 8.

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