If $\lim_{x \to 0} \frac{\sin x + ae^x + be^{-x} + c \ln(1 + x)}{x^3}$ is finite, find $a, b, c$ and the limit $L$:
(1) $a = -1$, $b = 1$, $c = 0$, $L = -1/3$
(2) $a = -1/2$, $b = 1/2$, $c = 0$, $L = -1/3$
(3) $a = 1/2$, $b = -1/2$, $c = 0$, $L = 1/3$
(4) $a = 1$, $b = -1$, $c = 0$, $L = 1/3$
Step-by-Step Solution
Key Concept: Expand in powers of $x$. Constant term: $a + b = 0$. $x$-coefficient: $1 + a - b + c = 0$. $x^2$-coefficient: $(a + b)/2 - c/2 = 0 \Rightarrow c = 0$. Solving: $a = -1/2$, $b = 1/2$. Then $L = (-1/6 + a/6 - b/6)/1 = -1/3$.
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Correct Answer: (2)