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If lim(x->infinity) [ (ax + 2) / sqrt(4 + x²) ]^( (ax³ + 1)/(x² + 1)) = L, where a >= 0 and L is a finite number, find the correct statement about a and L.

1. a = 0, L = 1
2. a = 1, L = e²
3. 0 < a < 1, L = 0
4. 0 <= a < 1, L = 0

Correct answer: 0 <= a < 1, L = 0

Solution

For large x: base = (ax+2)/sqrt(4+x²) ~ ax/x = a. Exponent ~ ax³/x² = ax. So expression ~ a^(ax). For L to be finite: if a > 1, a^(ax) -> infinity (not finite). If 0 < a < 1, a^(ax) -> 0 (finite). If a = 0, base = 2/sqrt(4+x²) -> 0, exponent = 1/(x²+1) -> 0; expression = 0⁰ -> need analysis: (2/sqrt(x²+4))^(1/(x²+1)). Taking log: [1/(x²+1)] * ln(2/sqrt(x²+4)) ~ [1/x²]*[ln2 - (1/2)ln(x²)] -> 0. So L = e⁰ = 1. Therefore a=0 gives L=1. For 0 < a < 1: L=0. For a=1: base->1, exponent->x, need detailed analysis. Combined: a=0 gives L=1 (option A), but option D says 0<=a<1, L=0 which contradicts a=0. The answer is option D only if a=0 also gives L=0, which it doesn't. Correct answers: a=0, L=1 (option A) AND 0<a<1 gives L=0.

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