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Let L = lim_(x -> inf) [ (a*x + 2) / sqrt(4 + x²) ]^(x² / (x+1)), where a >= 0 and L is a finite number. Which of the following is/are correct? (P) a = 0, L = 1 (Q) a = 1, L = e² (R) 0 < a < 1, L = 0 (S) 0 <= a < 1, L = 0

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

Correct answer: a = 1, L = e²

Solution

For a=1, (x+2)/sqrt(x²+4) -> 1 slowly; expand ln f(x) ~ 2/x, exponent ~ x, so L = e². For 0<=a<1, base->a<1 and exponent->inf, so L=0. For a>1, L=inf. Both Q and S (which includes a=0 giving L=0) are correct. But option S subsumes option R, and Q is the non-trivial finite case.

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