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\(lim\left(\sqrt[3]{n^3-2n^2}-2n+1\right)\)
\(=lim\dfrac{n^3-2n^2-\left(2n-1\right)^3}{\left(\sqrt[3]{n^3-2n^2}\right)^2+\sqrt[3]{n^3-2n^2}\left(2n-1\right)+\left(2n-1\right)^2}\)
\(=lim\dfrac{n^3-2n^2-\left(8n^3-12n^2+6n-1\right)}{\sqrt[3]{n^6-4n^5+4n^4}+\sqrt[3]{n^3-2n^2}\left(2n-1\right)+4n^2-4n+1}\)
\(=lim\dfrac{-7n^3+10n^2-6n+1}{\sqrt[3]{n^6-4n^5+4n^4}+\sqrt[3]{n^3-2n^2}\left(2n-1\right)+4n^2-4n+1}\)
\(=lim\dfrac{-7+\dfrac{10}{n}-\dfrac{6}{n^2}+\dfrac{1}{n^3}}{\sqrt[3]{\dfrac{1}{n^3}-\dfrac{4}{n^4}+\dfrac{4}{n^5}}+\sqrt[3]{\dfrac{1}{n^3}-\dfrac{2}{n^4}}\left(2-\dfrac{1}{n}\right)+\dfrac{4}{n}-\dfrac{4}{n^2}+\dfrac{1}{n^3}}\)
\(=\dfrac{-7}{0}=-\infty\)
\(\lim\limits\dfrac{n\left(\sqrt[3]{2-n^3}+n\right)}{\sqrt{n^2+1}-2n^2}\)
\(=lim\dfrac{n\cdot n\cdot\left(\sqrt[3]{\dfrac{2}{n^3}-1}+1\right)}{n^2\left(\sqrt{\dfrac{1}{n^2}+\dfrac{1}{n^4}}-2\right)}\)
\(=\lim\limits\dfrac{\sqrt[3]{\dfrac{2}{n^3}-1}+1}{\sqrt{\dfrac{1}{n^2}+\dfrac{1}{n^4}}-2}=\dfrac{\sqrt[3]{0-1}+1}{0-2}=\dfrac{-1+1}{-2}=0\)
\(\lim\limits_{x\rightarrow+\infty}\dfrac{x\sqrt{x}+x+1}{x^2+x+10}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{\dfrac{1}{\sqrt{x}}+\dfrac{1}{x}+\dfrac{1}{x^2}}{1+\dfrac{1}{x}+\dfrac{10}{x^2}}=\dfrac{0+0+0}{1+0+0}=0\)
