\(5,lim\left(\sqrt{n^2+n}-\sqrt{n^2+2}\right)\)
\(=lim\dfrac{\left(\sqrt{n^2+n}\right)^2-\left(\sqrt{n^2+2}\right)^2}{\left(\sqrt{n^2+n}\right)+\left(\sqrt{n^2+2}\right)^2}\)
\(=lim\dfrac{n^2+n-n^2-2}{\left(\sqrt{n^2+n}\right)+\left(\sqrt{n^2+2}\right)}\)
\(=lim\dfrac{n-2}{\left(\sqrt{n^2+n}\right)+\left(\sqrt{n^2+2}\right)}\)
\(=lim\dfrac{\dfrac{n}{n}-\dfrac{2}{n}}{\left(\dfrac{\sqrt{n^2}}{n^2}+\dfrac{\sqrt{n}}{n^2}\right)+\left(\dfrac{\sqrt{n^2}}{n^2}+\dfrac{2}{n^2}\right)}\)
\(=\dfrac{1-0}{1+0+1+0}\)
\(=\dfrac{1}{2}\)
\(7,lim\left(n+1-\sqrt{n^2+2n+5}\right)\)
\(=lim\left(n-\sqrt{n^2+2n+5}+1\right)\)
\(=lim\dfrac{\left(n-\sqrt{n^2+2n+5}\right)\left(n+\sqrt{n^2+2n+5}\right)}{n+\sqrt{n^2+2n+5}}+1\)
\(=lim\dfrac{n^2-n^2+2n+5}{n+\sqrt{n^2+2n+5}}+1\)
\(=lim\dfrac{2n+5}{n+\sqrt{n^2+2n+5}}+1\)
\(=lim\dfrac{\dfrac{2n}{n}+\dfrac{5}{n}}{\dfrac{n}{n}+\sqrt{\dfrac{n^2}{n^2}+\dfrac{2n}{n^2}+\dfrac{5}{n^2}}}+1\)
\(=lim\dfrac{2+\dfrac{5}{n}}{1+\sqrt{1+\dfrac{2}{n}+\dfrac{5}{n^2}}}+1\)
\(=\dfrac{2+0}{1+\sqrt{1+0+0}}+1\)
\(=\dfrac{2}{2}+1\)
\(=1+1\)
\(=2\)
\(12,lim\dfrac{\sqrt[3]{n^6-7n^3-5n+8}}{n+12}\)
\(=lim\dfrac{n^2.\sqrt[3]{\dfrac{n^6}{n^6}-\dfrac{7n^3}{n^6}-\dfrac{5n}{n^6}+\dfrac{8}{n^6}}}{n^2\left(\dfrac{n}{n^2}+\dfrac{12}{n^2}\right)}\)
\(=lim\dfrac{n^2.\sqrt[3]{1-\dfrac{7}{n^3}-\dfrac{5}{n^5}+\dfrac{8}{n^6}}}{n^2\left(\dfrac{1}{n}-\dfrac{12}{n^2}\right)}\)
\(=+\infty\)
vì \(\left\{{}\begin{matrix}lim n^2\left(\sqrt{1-\dfrac{7}{n^3}-\dfrac{5}{n^5}+\dfrac{8}{n^6}}\right)=1>0\\lim n^2\left(\dfrac{1}{n}+\dfrac{12}{n^2}\right)=0\end{matrix}\right.\)
7.
\(\lim \left( {n + 1 - \sqrt {{n^2} + 2n + 5} } \right) = \lim \left( {\frac{{{{\left( {n + 1} \right)}^2} - \left( {{n^2} + 2n + 5} \right)}}{{n + 1 + \sqrt {{n^2} + 2n + 5} }}} \right) = \lim \left( {\frac{{ - 4}}{{n + 1 + \sqrt {{n^2} + 2n + 5} }}} \right) = 0\)
