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

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

\(=\lim_{}\frac{4n^2+3}{\sqrt[3]{\left\lbrack n^3\left(8+\frac{4}{n}+\frac{1}{n^3}\right)^{}\right\rbrack^2}+\sqrt[3]{\left\lbrack n^3\left(8+\frac{4}{n}+\frac{1}{n^3}\right)\cdot n^3\cdot\left(8-\frac{2}{n^3}\right)\right\rbrack}+\sqrt[3]{\left\lbrack n^3\left(8-\frac{2}{n^3}\right)^2\right\rbrack}}\)

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

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

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

\(=\frac{4}{4+4+4}=\frac{4}{12}=\frac13\)

2: \(\lim_{}\left(\sqrt[3]{n^3+n^2+1}+\sqrt[3]{8-n^3}\right)\)

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

\(=\lim_{}\frac{n^2+9}{n^2\cdot\sqrt[3]{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)^2}-n^2\sqrt[3]{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)\left(\frac{8}{n^3}-1\right)}+n^2\cdot\sqrt[3]{\left(\frac{8}{n^3}-1\right)^2}}\)

\(=\lim_{}\frac{1+\frac{9}{n^2}}{\sqrt[3]{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)^2}-\sqrt[3]{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)\left(\frac{8}{n^3}-1\right)}+\sqrt[3]{\left(\frac{8}{n^3}-1\right)^2}}\)

\(=\frac{1+0}{\sqrt[3]{\left(1+0+0\right)^2}-\sqrt[3]{\left(1+0+0\right)\left(0-1\right)}+\sqrt[3]{\left(0-1\right)^2}}\)

\(=\frac{1}{1-1+1}=\frac11\) =1

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

\(=\lim_{}\frac{n^3+n^2+2-n^3}{\sqrt[3]{\left(n^3+n^2+2\right)^2}+n\cdot\sqrt[3]{n^3+n^2+2}+n^2}\)

\(=\lim_{}\frac{n^2+2}{n^2\cdot\sqrt[3]{\left(1+\frac{1}{n}+\frac{2}{n^3}\right)^2}+n^2\cdot\sqrt[3]{1+\frac{1}{n}+\frac{2}{n^3}}+n^2}\)

\(=\lim_{}\frac{1+\frac{2}{n^2}}{\sqrt[3]{\left(1+\frac{1}{n}+\frac{2}{n^3}\right)^2}+\sqrt[3]{1+\frac{1}{n}+\frac{2}{n^3}}+1}=\frac{1+0}{\sqrt1+1+1}=\frac13\)

1:

\(=\lim\limits_{n\rightarrow\infty}\dfrac{n^3+3n^2+1-n^3}{\sqrt[3]{n^3+3n^2+1}+n}\)

\(=\lim\limits_{n\rightarrow\infty}\dfrac{3n^2+1}{\sqrt[3]{n^3+3n^2+1}+n}\)

\(=\lim\limits_{n\rightarrow\infty}\dfrac{n^2\left(3+\dfrac{1}{n^2}\right)}{n\left(\sqrt[3]{1+\dfrac{3}{n}+\dfrac{1}{n^3}}+1\right)}\)

\(=\lim\limits_{n\rightarrow\infty}\dfrac{n\cdot\left(3+\dfrac{1}{n^2}\right)}{\sqrt[3]{1+\dfrac{3}{n}+\dfrac{1}{n^3}}+1}\)

\(=\lim\limits_{n\rightarrow\infty}n\cdot\lim\limits_{n\rightarrow\infty}\dfrac{3+\dfrac{1}{n^2}}{\sqrt[3]{1+\dfrac{3}{n}+\dfrac{1}{n^3}}+1}\)

\(=+\infty\) vì \(\left\{{}\begin{matrix}\lim\limits_{n\rightarrow\infty}n=+\infty\\\lim\limits_{n\rightarrow\infty}\dfrac{3+\dfrac{1}{n^2}}{\sqrt[3]{1+\dfrac{3}{n}+\dfrac{1}{n^3}}+1}=\dfrac{3}{2}>0\end{matrix}\right.\)

2: 

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

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

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

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

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

\(=0\)

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

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

\(=\lim_{}\left\lbrack n^2\cdot\left(\sqrt{1+\frac{2}{n^2}}\cdot\sqrt[3]{8+\frac{1}{n^3}}+\sqrt[2]{4+\frac{1}{n^2}}\cdot\sqrt[3]{1+\frac{2}{n^3}}\right)\right\rbrack\)

=+∞ vì \(\lim_{}n^2=\) +∞ và \(\lim_{}\left(\sqrt{1+\frac{2}{n^2}}\cdot\sqrt[3]{8+\frac{1}{n^3}}+\sqrt[2]{4+\frac{1}{n^2}}\cdot\sqrt[3]{1+\frac{2}{n^3}}\right)=1\cdot\sqrt[3]{8}+\sqrt[2]{4}\cdot\sqrt[3]{1}=1\cdot2+2\cdot1=4>0\)

a) lim \(\frac{\left(2n^2-3n+5\right)\left(2n+1\right)}{\left(4-3n\right)\left(2n^2+n+1\right)}\)

= lim \(\frac{\left(2-\frac{3}{n}+\frac{5}{n^2}\right)\left(2+\frac{1}{n}\right)}{\left(\frac{4}{n}-3\right)\left(2+\frac{1}{n}+\frac{1}{n^2}\right)}=\frac{4}{-6}=-\frac{2}{3}\)

b)lim ( \(\frac{\sqrt{n^4+1}}{n}-\frac{\sqrt{4n^6+2}}{n^2}\))

= lim ( \(\frac{n\sqrt{n^4+1}-\sqrt{4n^6+2}}{n^2}\) )

= lim \(\frac{\left(n^6+n^2\right)-\left(4n^6+2\right)}{n^2\left(n\sqrt{n^4+1}+\sqrt{4n^2+2}\right)}\)

= lim \(\frac{-3n^6+n^2+2}{n^3\sqrt{n^4+1}+n^2\sqrt{4n^2+2}}\)

= lim \(\frac{-3n\left(1-\frac{1}{n^4}-\frac{2}{n^6}\right)}{\sqrt{1+\frac{1}{n^4}}+\frac{1}{n^2}\sqrt{4+\frac{2}{n^2}}}\)

= lim \(-3n=-\infty\)

c) lim \(\frac{2n+3}{\sqrt{9n^2+3}-\sqrt[3]{2n^2-8n^3}}\)

= lim\(\frac{2+\frac{3}{n}}{\sqrt{9+\frac{3}{n^2}}-\sqrt[3]{\frac{2}{n}-8}}=\frac{2}{3+2}=\frac{2}{5}\)

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

\(=\lim\left(\dfrac{2n^3-13n^2+3}{\left(2n^2+3\right)\left(5n+1\right)}\right)=\lim\dfrac{2-\dfrac{13}{n}+\dfrac{3}{n^3}}{\left(2+\dfrac{3}{n^2}\right)\left(5+\dfrac{1}{n}\right)}=\dfrac{2}{2.5}=\dfrac{1}{5}\)

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

\(c=\lim\dfrac{\sqrt{1+\dfrac{3}{n^3}-\dfrac{2}{n^4}}}{2-\dfrac{2}{n}+\dfrac{3}{n^2}}=\dfrac{1}{2}\)

\(d=\lim\dfrac{\sqrt{1-\dfrac{4}{n}}-\sqrt{4+\dfrac{1}{n^2}}}{\sqrt{3+\dfrac{1}{n^2}}-1}=\dfrac{1-2}{\sqrt{3}-1}=-\dfrac{1+\sqrt{3}}{2}\)

Lim 3.4n-2.13n/5n+6.13n

a/ Bạn coi lại đề bài, 3n^2 +n^2 thì bằng 4n^2 luôn chứ ko ai cho đề bài như vậy cả

b/ \(\lim\limits\dfrac{\dfrac{n^3}{n^3}+\dfrac{3n}{n^3}+\dfrac{1}{n^3}}{-\dfrac{n^3}{n^3}+\dfrac{2n}{n^3}}=-1\)

c/ \(=\lim\limits\dfrac{-\dfrac{2n^3}{n^2}+\dfrac{3n}{n^2}+\dfrac{1}{n^2}}{-\dfrac{n^2}{n^2}+\dfrac{n}{n^2}}=\lim\limits\dfrac{-2n}{-1}=+\infty\)

d/ \(=\lim\limits\left[n\left(1+1\right)\right]=+\infty\)

e/ \(\lim\limits\left[2^n\left(\dfrac{2n}{2^n}-3+\dfrac{1}{2^n}\right)\right]=\lim\limits\left(-3.2^n\right)=-\infty\)

f/ \(=\lim\limits\dfrac{4n^2-n-4n^2}{\sqrt{4n^2-n}+2n}=\lim\limits\dfrac{-\dfrac{n}{n}}{\sqrt{\dfrac{4n^2}{n^2}-\dfrac{n}{n^2}}+\dfrac{2n}{n}}=-\dfrac{1}{2+2}=-\dfrac{1}{4}\)

g/ \(=\lim\limits\dfrac{n^2+3n-1-n^2}{\sqrt{n^2+3n-1}+n}+\lim\limits\dfrac{n^3-n^3+n}{\sqrt[3]{\left(n^3-n\right)^2}+n.\sqrt[3]{n^3-n}+n^2}\)

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

\(=\dfrac{3}{2}+0=\dfrac{3}{2}\)

a) lim \(\left(-3n^3+n^2-1\right)\)

minh le oi ban dao mau so cua ban len cho tu uong roi thay vi tri cua mau thanh n³ +2n

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

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

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

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

\(a=\lim\dfrac{1}{\sqrt{4n+1}+2\sqrt{n}}=\dfrac{1}{\infty}=0\)

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

\(c=\lim4^n\left(\sqrt{\left(\dfrac{9}{16}\right)^n-\left(\dfrac{3}{16}\right)^n}-1\right)=+\infty.\left(-1\right)=-\infty\)

\(d=\lim n^3\left(3+\dfrac{2}{n}+\dfrac{1}{n^2}\right)=+\infty.3=+\infty\)