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Consider the function f(x) defined as: f(x) = (e^x - e^(sin x)) / (a * x³) for x < 0; f(x) = b for x = 0; f(x) = x / ln(1 + 4x) for x > 0. If f is continuous at x = 0, find the value of (3a + 4b).

1. -3
2. 0
3. 3
4. 4

Correct answer: 3

Solution

The right-hand limit gives b = 1/4, and the left-hand limit of (e^x - e^(sin x))/(ax³) equals 1/(6a); setting 1/(6a) = 1/4 gives a = 2/3, so 3a + 4b = 2 + 1 = 3.

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