A

A

\(7,lim\left(n+1-\sqrt{n^2+2n+5}\right)\)

\(=lim\left(n-\sqrt{n^2+2n+5}+1\right)\)

\(=lim\dfrac{\left(n-\sqrt{n^2+2n+5}\right)\left(n+\sqrt{n^2+2n+5}\right)}{n+\sqrt{n^2+2n+5}}+1\)

\(=lim\dfrac{n^2-n^2+2n+5}{n+\sqrt{n^2+2n+5}}+1\)

\(=lim\dfrac{2n+5}{n+\sqrt{n^2+2n+5}}+1\)

\(=lim\dfrac{\dfrac{2n}{n}+\dfrac{5}{n}}{\dfrac{n}{n}+\sqrt{\dfrac{n^2}{n^2}+\dfrac{2n}{n^2}+\dfrac{5}{n^2}}}+1\)

\(=lim\dfrac{2+\dfrac{5}{n}}{1+\sqrt{1+\dfrac{2}{n}+\dfrac{5}{n^2}}}+1\)

\(=\dfrac{2+0}{1+\sqrt{1+0+0}}+1\)

\(=\dfrac{2}{2}+1\)

\(=1+1\)

\(=2\)

\(5,lim\left(\sqrt{n^2+n}-\sqrt{n^2+2}\right)\)

\(=lim\dfrac{\left(\sqrt{n^2+n}\right)^2-\left(\sqrt{n^2+2}\right)^2}{\left(\sqrt{n^2+n}\right)+\left(\sqrt{n^2+2}\right)^2}\)

\(=lim\dfrac{n^2+n-n^2-2}{\left(\sqrt{n^2+n}\right)+\left(\sqrt{n^2+2}\right)}\)

\(=lim\dfrac{n-2}{\left(\sqrt{n^2+n}\right)+\left(\sqrt{n^2+2}\right)}\)

\(=lim\dfrac{\dfrac{n}{n}-\dfrac{2}{n}}{\left(\dfrac{\sqrt{n^2}}{n^2}+\dfrac{\sqrt{n}}{n^2}\right)+\left(\dfrac{\sqrt{n^2}}{n^2}+\dfrac{2}{n^2}\right)}\)

\(=\dfrac{1-0}{1+0+1+0}\)

\(=\dfrac{1}{2}\)

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