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Evaluate: lim_(x->0) [cbrt(1 - arctan(2x)) - cbrt(1 + arcsin(2x))] / [sqrt(1 - arcsin(3x)) - sqrt(1 + arctan(3x))]

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

Correct answer: 4/9

Solution

Applying first-order expansions, the numerator reduces to -(4/3)x and the denominator reduces to -3x, giving the limit 4/9.

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