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Consider f(x) = lim(n->inf) [sgn(sqrt(ac) - b)*e^(nx) + x² + f] / [2*e^(nx) + x + d], where a > b > c > 0 and d, f are real numbers. Here sgn(y) is the signum function. Given that a, b, c are in arithmetic progression and f(x) is continuous for all real x, find the value of (2f + d + 1).

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

Correct answer: 1

Solution

Since a,b,c in AP: b = (a+c)/2 = AM(a,c) >= sqrt(ac) = GM(a,c), so sqrt(ac) - b <= 0 (strict inequality when a != c). Thus sgn(sqrt(ac)-b) = -1. For x>0: limit = -1/2 (e^(nx) dominates). For x<0: limit = (x²+f)/(x+d). Continuity at x=0 requires (0+f)/(0+d) = f/d = -1/2, so f = -d/2. Then 2f + d + 1 = -d + d + 1 = 1.

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