\(lim\left(2n-\sqrt[3]{8n^3+8n^2+2}\right)\)

\(=lim\frac{\left(2n-\sqrt[3]{8n^3+8n^2+2}\right)\left(4n^2+2n\sqrt[3]{8n^3+8n^2+2}+\sqrt[3]{\left(8n^3+8n^2+2\right)^2}\right)}{4n^2+2n\sqrt[3]{8n^3+8n^2+2}+\sqrt[3]{\left(8n^3+8n^2+2\right)^2}}\)

\(=lim\frac{8n^3-\left(8n^3+8n^2+2\right)}{4n^2+2n\sqrt[3]{8n^3+8n^2+2}+\sqrt[3]{\left(8n^3+8n^2+2\right)^2}}\)

\(=lim\frac{-8n^2-2}{4n^2+2n\sqrt[3]{8n^3+8n^2+2}+\sqrt[3]{\left(8n^3+8n^2+2\right)^2}}\)

\(=lim\frac{-8-\frac{2}{n^2}}{4+2\sqrt[3]{8+\frac{8}{n}+\frac{2}{n^3}}+\sqrt[3]{\left(8+\frac{8}{n}+\frac{2}{n^3}\right)^2}}\)

\(=\frac{-8+0}{4+2\sqrt[3]{8+8+0}+\sqrt[3]{\left(8+0+0\right)^2}}=\frac{-2}{3}\)

\(=\lim\left(\sqrt[]{4n^2+2n+1}-2n+2n-\sqrt[3]{8n^3-3n^2+1}\right)\)

\(=\lim\left(\dfrac{2n+1}{\sqrt[]{4n^2+2n+1}+2n}+\dfrac{3n^2-1}{4n^2+2n\sqrt[3]{8n^3-3n^2+1}+\sqrt[3]{\left(8n^3-3n^2+1\right)^2}}\right)\)

\(=\lim\left(\dfrac{2+\dfrac{1}{n}}{\sqrt[]{4+\dfrac{2}{n}+\dfrac{1}{n^2}}+2}+\dfrac{3-\dfrac{1}{n^2}}{4+2\sqrt[3]{8-\dfrac{3}{n}+\dfrac{1}{n^3}}+\sqrt[3]{\left(8-\dfrac{3}{n}+\dfrac{1}{n^3}\right)^2}}\right)\)

\(=\dfrac{2}{\sqrt[]{4}+2}+\dfrac{3}{4+2\sqrt[3]{8}+\sqrt[3]{8^2}}=...\)

\(=\lim\dfrac{\sqrt{4-\dfrac{1}{n}}+\sqrt[3]{8+\dfrac{1}{n}}}{2+\dfrac{3}{n}}=\dfrac{2+2}{2}=2\)

\(\lim\limits\left(2-3n\right)^4\left(n+1\right)^3=\lim n^7\left(3-\dfrac{2}{n}\right)^4\left(1+\dfrac{1}{n}\right)^3=+\infty\)

\(\lim\left(\sqrt[3]{n+4}-\sqrt[3]{n+1}\right)=\lim\dfrac{3}{\sqrt[3]{\left(n+4\right)^2}+\sqrt[3]{\left(n+4\right)\left(n+1\right)}+\sqrt[3]{\left(n+1\right)^2}}=0\)

\(\lim\left(\sqrt[3]{8n^3+3n^2+4}-2n+6\right)=\lim\dfrac{8n^3+3n^2+4-\left(2n-6\right)^3}{\sqrt[3]{\left(8n^3+3n^2+4\right)^2}+\left(2n-6\right)\sqrt[3]{8n^3+3n^2+4}+\left(2n-6\right)^2}\)

\(=\lim\dfrac{75n^2-216n+220}{\sqrt[3]{\left(8n^3+3n^2+4\right)^2}+\left(2n-6\right)\sqrt[3]{8n^3+3n^2+4}+\left(2n-6\right)^2}\)

\(=\lim\dfrac{75-\dfrac{216}{n}+\dfrac{220}{n^2}}{\sqrt[3]{\left(8+\dfrac{3}{n}+\dfrac{4}{n^3}\right)^2}+\left(2-\dfrac{6}{n}\right)\sqrt[3]{8+\dfrac{3}{n}+\dfrac{4}{n^3}}+\left(2-\dfrac{6}{n}\right)^2}\)

\(=\dfrac{75}{\sqrt[3]{8^2}+2.\sqrt[3]{8}+2^2}=...\)

d.

\(\lim\left(\sqrt[3]{8n^3+3n^2-2}+\sqrt[3]{5n^2-8n^3}\right)\)

\(=\lim\left(\sqrt[3]{8n^3+3n^2-2}-\sqrt[3]{8n^3-5n^2}\right)\)

\(=\lim\dfrac{8n^3+3n^2-2-\left(8n^3-5n^2\right)}{\sqrt[3]{\left(8n^3+3n^2-2\right)^2}+\sqrt[3]{\left(8n^3+3n^2-2\right)\left(8n^3-5n^2\right)}+\sqrt[3]{8n^3-5n^2}}\)

\(=\lim\dfrac{8n^2-2}{\sqrt[3]{\left(8n^3+3n^2-2\right)^2}+\sqrt[3]{\left(8n^3+3n^2-2\right)\left(8n^3-5n^2\right)}+\sqrt[3]{8n^3-5n^2}}\)

\(=lim\dfrac{8-\dfrac{2}{n^2}}{\sqrt[3]{\left(8+\dfrac{3}{n}-\dfrac{2}{n^3}\right)^2}+\sqrt[3]{\left(8+\dfrac{3}{n}-\dfrac{2}{n^3}\right)\left(8-\dfrac{5}{n}\right)}+\sqrt[3]{\left(8-\dfrac{5}{n}\right)^2}}\)

\(=\dfrac{8}{\sqrt[3]{8^2}+\sqrt[3]{8.8}+\sqrt[3]{8^2}}=...\)

a. ĐKXĐ: \(n\ge0\)

\(lim_{n\rightarrow0}\dfrac{\sqrt{2n+1}}{\sqrt{8n}+1}=\dfrac{\sqrt{2.0+1}}{\sqrt{8.0}+1}=1\)

\(lim_{n\rightarrow+\infty}\dfrac{\sqrt{2n+1}}{\sqrt{8n}+1}=lim_{n\rightarrow+\infty}\dfrac{\sqrt{2+\dfrac{1}{n}}}{\sqrt{8}+\dfrac{1}{\sqrt{n}}}=\dfrac{1}{2}\)

b. ĐKXĐ: \(\left\{{}\begin{matrix}n\ne0\\n\le\dfrac{-1-\sqrt{21}}{2}\\n\ge\dfrac{-1+\sqrt{21}}{2}\end{matrix}\right.\)

\(lim_{n\rightarrow+\infty}\dfrac{3n+\sqrt{n^2+n-5}}{-2n}=\)\(lim_{n\rightarrow+\infty}\dfrac{3+\sqrt{1+\dfrac{1}{n}-\dfrac{5}{n^2}}}{-2}=-2\)

\(lim_{n\rightarrow-\infty}\dfrac{3n+\sqrt{n^2+n-5}}{-2n}=\)\(lim_{n\rightarrow-\infty}\dfrac{3+\sqrt{1+\dfrac{1}{n}-\dfrac{5}{n^2}}}{-2}=-1\)

a, \(lim\dfrac{\sqrt{2n+1}}{\sqrt{8n}+1}=lim\dfrac{\sqrt{n}.\sqrt{2+\dfrac{1}{n}}}{\sqrt{n}\left(\sqrt{8}+\dfrac{1}{n}\right)}=\dfrac{\sqrt{2}}{\sqrt{8}}=\dfrac{1}{2}\)

b, \(lim\dfrac{3n+\sqrt{n^2+n-5}}{-2n}\)

\(=lim\left(\dfrac{3}{2}-\dfrac{\sqrt{n^2+n-5}}{2n}\right)\)

\(=lim\left(\dfrac{3}{2}-\dfrac{n\sqrt{1+\dfrac{1}{n}-\dfrac{5}{n^2}}}{2n}\right)=\dfrac{3}{2}-\dfrac{1}{2}=1\)

\(\lim\dfrac{\sqrt{2n+1}}{\sqrt{8n}+1}=\lim\dfrac{\sqrt{n}.\sqrt{2+\dfrac{1}{n}}}{\sqrt{n}\left(\sqrt{8}+\dfrac{1}{\sqrt{n}}\right)}=\lim\dfrac{\sqrt{2+\dfrac{1}{n}}}{\sqrt{8}+\dfrac{1}{\sqrt{n}}}=\dfrac{\sqrt{2}}{\sqrt{8}}=\dfrac{1}{2}\)

\(\lim\dfrac{3n+\sqrt{n^2+n-5}}{-2n}=\lim\dfrac{n\left(3+\sqrt{1+\dfrac{1}{n}-\dfrac{5}{n^2}}\right)}{-2n}=\lim\dfrac{3+\sqrt{1+\dfrac{1}{n}-\dfrac{5}{n^2}}}{-2}=\dfrac{3+1}{-2}=-2\)

1) = lim n. \(\frac{n^3-3n^2-27n^3}{\sqrt[3]{\left(n^3-3n^2\right)^2}+3n\sqrt[3]{n^3-3n^2}+9n^2}\)

= lim \(\frac{n\left(-26n^3-3n^2\right)}{\sqrt[3]{\left(n^3-3n^2\right)^2}+3n\sqrt[3]{n^3-3n^2}+9n^2}\)

= lim \(\frac{n^2\left(-26-\frac{3}{n}\right)}{\sqrt[3]{\left(1-\frac{3}{n}\right)^2}+3\sqrt[3]{1-\frac{3}{n}}+9}\)

= lim \(\frac{n^2\left(-26\right)}{13}=-\infty\)

2) = lim ( \(\sqrt{4n^2+n}-2n+\sqrt[3]{2n^2-8n^3}+2n\))

= lim ( \(\frac{n}{\sqrt{4n^2+n}+2n}+\frac{2n^2}{\sqrt[3]{\left(2n^2-8n^3\right)^2}-2n\sqrt[3]{2n^2-8n^3}+4n^2}\))

= \(\frac{1}{2+2}+\frac{2}{4+4+4}=\frac{5}{12}\)