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If lim_(x->inf) [ (a*x + 2) / sqrt(4 + x²) ]^((x²+3)/(x²+1)) = L, where a >= 0 and L is a finite number, find the values of 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

Interpreting the exponent as x*(x²+3)/(x²+1) ~ x as x->inf: base -> a. If 0<=a<1: a^x -> 0. If a=1: 1^x = 1 (finite but exponent not needed). If a>1: a^x -> inf (not finite). So for L to be finite with a>=0, we need 0<=a<1, giving L=0. Answer: option D.

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