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Determine the value of L if L = lim x→0 [(sin x + ae^x + be^−x + c ln(1 + x)) / x³], given that L is finite and not infinite.

1. 1/2
2. −1/3
3. −1/6
4. 3

Correct answer: −1/3

Solution

Expanding gives a+b=0, 1+a-b+c=0, and a+b-c=0 so c=0, a=-1/2, b=1/2. The x^3 coefficient is -1/6 + (a-b)/6 = -1/6 - 1/6 = -1/3, so L = -1/3 (idx 1). The stored idx 2 (-1/6) is wrong.

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