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Given that lim_(x -> 0) (a*tan(x) + b*x + c*x² + x³) / (2*x²*(e^x - 1) - 2*x³ - x⁴) = d, where d is a finite nonzero value, determine which of the following are correct.

1. a = 3
2. b = 3
3. c = 0
4. d = 2/5

Correct answer: b = 3

Solution

Denominator expansion: 2x²*(e^x-1) - 2x³ - x⁴ = 2x²*(x + x²/2 + x³/6 + x⁴/24 +...) - 2x³ - x⁴ = 2x³ + x⁴ + x⁵/3 +... - 2x³ - x⁴ = x⁵/3 + O(x⁶). So denominator ~ x⁵/3. For finite limit, numerator must also be O(x⁵). Numerator: a*tan(x) + b*x + c*x² + x³ = a*(x + x³/3 + 2x⁵/15 +...) + b*x + c*x² + x³ = (a+b)*x + c*x² + (1+a/3)*x³ +.... For O(x⁵) start: coefficient of x must be 0 => a+b = 0; coefficient of x² must be 0 => c = 0; coefficient of x³ must be 0 => 1 + a/3 = 0 => a = -3; then b = 3; coefficient of x⁵ term of numerator = 2a/15 = -6/15 = -2/5. Limit = (-2/5)/(1/3) = -6/5. Hmm. Let me recheck: d = (2a/15)/(1/3) = 6a/15 = 2a/5. With a = -3: d = -6/5. Answer: a = -3, b = 3, c = 0, d = -6/5. So correct options: b = 3 and c = 0.

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