JEEOlympiad
Compute L = lim_{x->infinity} ( (x^3 + 3 x^2)^(1/3) - (x^2 - 2…
Question
Compute L = lim_{x->infinity} ( (x^3 + 3 x^2)^(1/3) - (x^2 - 2 x)^(1/2) ). Give L.
✓ Verified answer: 2checked by our engine — not a guess
Step-by-step solution
(x^3 + 3x^2)^(1/3) = x·(1 + 3/x)^(1/3) = x·(1 + (1/3)(3/x) + O(1/x^2)) = x + 1 + O(1/x).
(x^2 - 2x)^(1/2) = x·(1 - 2/x)^(1/2) = x·(1 + (1/2)(-2/x) + O(1/x^2)) = x - 1 + O(1/x).
Subtracting: (x+1) - (x-1) + O(1/x) = 2 + O(1/x) -> 2. Hence L = 2.
Final answer2
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