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Let f(x) = lim_(n->inf) (cos(sqrt(x/n)))ⁿ and g(x) = lim_(n->inf) (1 - x + x/sqrt(e))ⁿ. The value of lim_(x->0+) [ln(f(x)) / ln(g(x))] is:

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

Correct answer: 0

Solution

f(x) = exp(lim_(n->inf) n * ln(cos(sqrt(x/n)))). Since cos(u) ≈ 1 - u²/2 for small u, ln(cos u) ≈ -u²/2. With u = sqrt(x/n): n * (-x/(2n)) = -x/2. So f(x) = e^(-x/2) and ln f(x) = -x/2 -> 0 as x -> 0+. For g(x): the base equals 1 - x*(1 - e^(-1/2)), a constant less than 1 for x > 0. Taking it to the power n -> inf gives g(x) = 0, so ln g(x) = -inf. Thus the ratio = (-x/2)/(-inf) = 0.

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