Вычислим A=\lim \limits_{n\to +\infty } (\sqrt{4n^2+6n -1}-\sqrt{4n^2+2n+5}). \sqrt{4n^2+6n-1}-\sqrt{4n^2+2n+5}=\dfrac{(\sqrt{4n^2+6n-1}-\sqrt{4n^2+2n+5})(\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})}{\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5}}=\dfrac{(4n^2+6n-1)-(4n^2+2n+5)}{\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5} }=\dfrac{4n-6}{\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5}}=\dfrac{4-\dfrac{6}{n}}{\sqrt{4+\dfrac{6}{n}-\dfrac{1}{n^2}}+\sqrt{4+\dfrac{2}{n}+\dfrac{5}{n^2}}}. ((\sqrt{4n^2+6n-1}-\sqrt{4n^2+2n+5})(\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})) / (\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})

Задание

Вычислим \(A=\lim \limits\_{n\to +\infty } (\sqrt{4n^2+6n -1}-\sqrt{4n^2+2n+5})\) .

\(\sqrt{4n^2+6n-1}-\sqrt{4n^2+2n+5}=\dfrac{(\sqrt{4n^2+6n-1}-\sqrt{4n^2+2n+5})(\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})}{\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5}}=\dfrac{(4n^2+6n-1)-(4n^2+2n+5)}{\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5} }=\dfrac{4n-6}{\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5}}=\dfrac{4-\dfrac{6}{n}}{\sqrt{4+\dfrac{6}{n}-\dfrac{1}{n^2}}+\sqrt{4+\dfrac{2}{n}+\dfrac{5}{n^2}}}\) .

\(((\sqrt{4n^2+6n-1}-\sqrt{4n^2+2n+5})(\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})) / (\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})\)

[ \(((\sqrt{4n^2+6n-1}-\sqrt{4n^2+2n+5})(\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})) / (\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})\) | \(((\sqrt{4n^2+6n-1}-\sqrt{4n^2+2n+5})(\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})) / (\sqrt{4n^2+6n-1}+\sqrt{4n^2+2n+5})\) |123]

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