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lim-x-3-x-1-3-x-0-

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Question Number 216698 by sniper237 last updated on 16/Feb/25
lim_(x→+∞) ^3 (√(x+1)) −^3 (√x) =^? 0
limx→+∞3x+1−3x=?0
Answered by MrGaster last updated on 16/Feb/25
((x+1))^(1/3) −^3 (√x)=(x+1)^(1/3) −x^(1/3) (x+1)^(1/3) =x^(1/3) +(1/3)x^((1/3)−1) +O(x^((1/3)−2) ) (x+1)^(1/3) −x^(1/3) =x^(1/3) +(1/3)x^(−(2/3)) +O(x^(−(5/3)) )−x^(1/3) (1/3)x^(−(2/3)) +O(x^(−(5/3)) ) lim_(x→∞^+ ) ((1/3)x^(−(2/3)) +O(x^(−(5/3)) ))=0
x+13−3x=(x+1)13−x13(x+1)13=x13+13x13−1+O(x13−2)(x+1)13−x13=x13+13x−23+O(x−53)−x1313x−23+O(x−53)limx→∞+(13x−23+O(x−53))=0
Answered by mehdee7396 last updated on 17/Feb/25
(((x+1))^(1/3) −(x)^(1/3) )=(1/( (((x+1)^2 ))^(1/3) +(((x+1)x))^(1/3) +(x^2 )^(1/3) )) ⇒lim_(x→∞) (((x+1))^(1/3) −(x)^(1/3) )=0
(x+13−x3)=1(x+1)23+(x+1)x3+x23⇒limx→∞(x+13−x3)=0

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