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