a: \(=\lim\limits\dfrac{n^2-n-n^2}{\sqrt{n^2-n}+n}\)
\(=\lim\limits\dfrac{-n}{\sqrt{n^2-n}+n}\)
\(=\lim\limits\dfrac{-1}{\sqrt{1-\dfrac{1}{n}}+1}=\dfrac{-1}{2}\)
b: \(=\lim\limits\dfrac{n-n^3+n^3}{\sqrt[3]{\left(n-n^3\right)^2}-n\sqrt[3]{n-n^3}+\sqrt[3]{n^2}}+2\)
\(=lim\left(\dfrac{n}{\sqrt[3]{\left(n-n^3\right)^2}-n\sqrt[3]{n-n^3}+\sqrt[3]{n^2}}\right)+2\)
\(=lim\left(\dfrac{\dfrac{1}{n}}{\sqrt[3]{\left(\dfrac{1}{n^2}-1\right)^2}-\dfrac{1}{n^4}\cdot\sqrt[3]{\dfrac{1}{n^2}-1}+\sqrt[3]{\dfrac{1}{n^4}}}\right)+2\)
\(=0+2=2\)
c: \(=lim\left(\sqrt{n+5}\cdot\dfrac{2n+3-2n+1}{\sqrt{2n+3}+\sqrt{2n-1}}\right)\)
\(=lim\left(\dfrac{2\sqrt{n+5}}{\sqrt{2n+3}+\sqrt{2n-1}}\right)\)
\(=lim\left(2\cdot\dfrac{\sqrt{1+\dfrac{5}{n}}}{\sqrt{2+\dfrac{3}{n}}+\sqrt{2-\dfrac{1}{n}}}\right)=2\cdot\dfrac{1}{2\sqrt{2}}=\dfrac{\sqrt{2}}{2}\)
d: \(=\lim\limits\dfrac{n^3+2n^2+1-n^3}{\sqrt[3]{\left(n^3+2n^2+1\right)^2}+n\sqrt[3]{n^3+2n^2+1}+n^2}\)
\(=\lim\limits\dfrac{2n^2+1}{\sqrt[3]{\left(n^3+2n^2+1\right)^2}+n\sqrt[3]{n^3+2n^2+1}+n^2}\)
\(=\lim\limits\dfrac{2+\dfrac{1}{n^2}}{\sqrt[3]{\left(1+\dfrac{2}{n}+\dfrac{1}{n^3}\right)^2+\sqrt[3]{1+\dfrac{2}{n}+\dfrac{1}{n^3}}+1}}=\dfrac{2}{3}\)
