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If beta = lim(x->0) [ e^(x³) - (1 - x³)^(1/3) + ((1 - x²)^(1/2) - 1) * sin(x) ] / [ x * sin²(x) ], then find the value of (18/5) * beta.

1. 3
2. 4
3. 5
4. 11

Correct answer: 5

Solution

Expanding numerator to leading order x³: e^(x³) - (1-x³)^(1/3) ~ (1+x³) - (1 - x³/3) = x³ + x³/3 = 4x³/3. The ((1-x²)^(1/2) - 1)*sin(x) ~ (-x²/2)*x = -x³/2. Total numerator ~ (4/3 - 1/2)x³ = (8/6 - 3/6)x³ = (5/6)x³. Denominator x*sin²(x) ~ x³. So beta = 5/6, and (18/5)*beta = (18/5)*(5/6) = 3. Wait - re-checking: beta = 5/6, 18/5 * 5/6 = 18/6 = 3. But option says 5. Let me recompute: (1-x³)^(1/3) = 1 - x³/3 +... so e^(x³) - (1-x³)^(1/3) = (1+x³+x⁶/2+...) - (1-x³/3-...) = x³ + x³/3 = 4x³/3. And ((1-x²)^(1/2)-1)sinx = (-x²/2 - x⁴/8 -...)(x - x³/6+...) = -x³/2 +... Numerator = 4x³/3 - x³/2 = (8-3)x³/6 = 5x³/6. Beta = 5/6. (18/5)*(5/6) = 3. Answer is 3.

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