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If lim_{x→0} [αx e^x − β log_e(1+x) + γx^2 e^{-x}] / [x sin^2 x] = 10, α, β, γ ∈ R then the value of α + β + γ is _____.

1. 1
2. 2
3. 3
4. 4

Correct answer: 1

Solution

To evaluate the limit as x approaches 0, we need to analyze the behavior of the numerator and denominator. By applying L'Hôpital's rule or Taylor series expansion, we find that the coefficients of the leading terms must satisfy a specific relationship to yield the limit of 10, leading to the conclusion that α + β + γ equals 1.

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