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NV
Nguyễn Việt Lâm
Giáo viên
18 tháng 1 2022
7/
\(=\lim\dfrac{n^2+4n+1-n^2}{\sqrt{n^2+4n+1}+n}=\lim\dfrac{4n+1}{\sqrt{n^2+4n+1}+n}=\lim\dfrac{4+\dfrac{1}{n}}{\sqrt{1+\dfrac{4}{n}+\dfrac{1}{n^2}}+1}=\dfrac{4}{1+1}=2\)
8/
\(=\lim\dfrac{n^2-\left(n^2+9n-1\right)}{n+\sqrt{n^2+9n-1}}=\lim\dfrac{-9n+1}{n+\sqrt{n^2+9n-1}}=\lim\dfrac{-9+\dfrac{1}{n}}{1+\sqrt{1+\dfrac{9}{n}-\dfrac{1}{n^2}}}=\dfrac{-9}{1+1}=-\dfrac{9}{2}\)
9/
Do \(1+2+...+n=\dfrac{n\left(n+1\right)}{2}=\dfrac{n^2+n}{2}\)
\(\Rightarrow\lim\dfrac{1+2+...+n}{n^2-1}=\lim\dfrac{n^2+n}{2n^2-2}=\lim\dfrac{1+\dfrac{1}{n}}{2-\dfrac{2}{n^2}}=\dfrac{1}{2}\)
NV
Nguyễn Việt Lâm
Giáo viên
13 tháng 1 2021
\(a=\lim\dfrac{\dfrac{1}{n}+\dfrac{1}{n^2}}{1+\dfrac{2}{n}}=\dfrac{0}{1}=0\)
\(b=\lim n^3\left(-2+\dfrac{1}{n}+\dfrac{2}{n^3}\right)=+\infty.\left(-2\right)=-\infty\)
\(c=\lim\dfrac{\sqrt{9-\dfrac{1}{n}-\dfrac{1}{n^2}}}{4-\dfrac{2}{n}}=\dfrac{\sqrt{9}}{4}=\dfrac{3}{4}\)
\(d=\lim\dfrac{\left(\dfrac{3}{4}\right)^n+5}{1+\left(\dfrac{2}{4}\right)^n}=\dfrac{5}{1}=5\)
NV
Nguyễn Việt Lâm
Giáo viên
17 tháng 1 2021
\(a=\lim\left(\dfrac{2n^3\left(5n+1\right)+\left(2n^2+3\right)\left(1-5n^2\right)}{\left(2n^2+3\right)\left(5n+1\right)}\right)\)
\(=\lim\left(\dfrac{2n^3-13n^2+3}{\left(2n^2+3\right)\left(5n+1\right)}\right)=\lim\dfrac{2-\dfrac{13}{n}+\dfrac{3}{n^3}}{\left(2+\dfrac{3}{n^2}\right)\left(5+\dfrac{1}{n}\right)}=\dfrac{2}{2.5}=\dfrac{1}{5}\)
\(b=\lim\left(\dfrac{n-2}{\sqrt{n^2+n}+\sqrt{n^2+2}}\right)=\lim\dfrac{1-\dfrac{2}{n}}{\sqrt{1+\dfrac{1}{n}}+\sqrt{1+\dfrac{2}{n}}}=\dfrac{1}{2}\)
\(c=\lim\dfrac{\sqrt{1+\dfrac{3}{n^3}-\dfrac{2}{n^4}}}{2-\dfrac{2}{n}+\dfrac{3}{n^2}}=\dfrac{1}{2}\)
\(d=\lim\dfrac{\sqrt{1-\dfrac{4}{n}}-\sqrt{4+\dfrac{1}{n^2}}}{\sqrt{3+\dfrac{1}{n^2}}-1}=\dfrac{1-2}{\sqrt{3}-1}=-\dfrac{1+\sqrt{3}}{2}\)
20 tháng 11 2023
\(\lim\limits\dfrac{\sqrt{4n^2+1}+2n-1}{\sqrt{n^2+4n+1}+n}\)
\(=\lim\limits\dfrac{\sqrt{4+\dfrac{1}{n^2}}+2-\dfrac{1}{n}}{\sqrt{1+\dfrac{4}{n}+\dfrac{1}{n^2}}+1}=\dfrac{2+2}{1+1}=\dfrac{4}{2}=2\)
\(\lim\limits\left[\sqrt{n}\left(\sqrt{n+1}-n\right)\right]\)
\(=\lim\limits\left[\sqrt{n^2+n}-\sqrt{n^3}\right]\)
\(=\lim\limits\dfrac{n^2+n-n^3}{\sqrt{n^2+n}+\sqrt{n^3}}\)
\(=\lim\limits\dfrac{n^3\left(-1+\dfrac{1}{n}+\dfrac{1}{n^2}\right)}{\sqrt{n^3\left(\dfrac{1}{n}+\dfrac{1}{n^2}\right)}+\sqrt{n^3}}\)
\(=\lim\limits\dfrac{n^3\left(-1+\dfrac{1}{n}+\dfrac{1}{n^2}\right)}{\sqrt{n^3}\left(\sqrt{\dfrac{1}{n}+\dfrac{1}{n^2}}+1\right)}\)
\(=\lim\limits\dfrac{n\sqrt{n}\left(-1+\dfrac{1}{n}+\dfrac{1}{n^2}\right)}{\sqrt{\dfrac{1}{n}+\dfrac{1}{n^2}}+1}\)
\(=-\infty\) vì \(\left\{{}\begin{matrix}lim\left(n\sqrt{n}\right)=+\infty\\lim\left(\dfrac{-1+\dfrac{1}{n}+\dfrac{1}{n^2}}{\sqrt{\dfrac{1}{n}+\dfrac{1}{n^2}}+1}\right)=-\dfrac{1}{1}=-1< 0\end{matrix}\right.\)
NV
Nguyễn Việt Lâm
Giáo viên
18 tháng 1 2022
1/...
2/ \(=\lim\dfrac{\dfrac{1}{n\sqrt{n}}-1}{4+\dfrac{1}{n^2\sqrt{n}}}=\dfrac{0-1}{4+0}=-\dfrac{1}{4}\) (chia cả tử-mẫu cho \(n^3\))
3/ \(=\lim\dfrac{3-\left(\dfrac{1}{4}\right)^n}{2.\left(\dfrac{3}{4}\right)^n+4\left(\dfrac{1}{4}\right)^n}=\dfrac{3-0}{2.0+3.0}=\dfrac{3}{0}=+\infty\) (chia tử mẫu cho \(4^n\))
4/ \(=\lim\dfrac{2.2^n+\dfrac{4}{3}.3^n}{1-\dfrac{1}{2}.2^n+3.3^n}=\lim\dfrac{2.\left(\dfrac{2}{3}\right)^n+\dfrac{4}{3}}{\left(\dfrac{1}{3}\right)^n-\dfrac{1}{2}.\left(\dfrac{2}{3}\right)^n+3}=\dfrac{2.0+\dfrac{4}{3}}{0-\dfrac{1}{2}.0+3}=\dfrac{4}{9}\) (chia tử mẫu cho \(3^n\))
PH
25 tháng 2 2020
lim\(\frac{3n^2+n-5}{2n^2+1}\)=lim\(\frac{n^2\left(3+\frac{1}{n}-\frac{5}{n^2}\right)}{n^2\left(2+\frac{1}{n}\right)}\)=\(\frac{3}{2}\)
PH
25 tháng 2 2020
lim\(\frac{\sqrt{9n^2-n}+1}{4n-2}\)=lim\(\frac{n\sqrt{9-\frac{1}{n}+\frac{1}{n^2}}}{n\left(4-\frac{2}{n}\right)}\)=lim\(\frac{\sqrt{9}}{4}\)=\(\frac{3}{2}\)
NV
Nguyễn Việt Lâm
Giáo viên
18 tháng 2 2020
\(lim\frac{\sqrt{9n^2+2n}+n-2}{\sqrt{4n^2+1}}=lim\frac{\sqrt{9+\frac{2}{n}}+1-\frac{2}{n}}{\sqrt{4+\frac{1}{n^2}}}=\frac{\sqrt{9}+1}{\sqrt{4}}=2\)
\(lim\frac{n}{\sqrt{4n^2+2}+\sqrt{n^2}}=lim\frac{1}{\sqrt{4+\frac{2}{n^2}}+\sqrt{1}}=\frac{1}{\sqrt{4}+\sqrt{1}}=\frac{1}{3}\)
\(lim\frac{\sqrt{4n+2}-\sqrt{2n-5}}{\sqrt{n+3}}=lim\frac{\sqrt{4+\frac{2}{n}}-\sqrt{2-\frac{5}{n}}}{\sqrt{1+\frac{3}{n}}}=\frac{2-\sqrt{2}}{1}=2-\sqrt{2}\)
l\\(lim\frac{\sqrt{4n^2+n+1}-n}{n^2+2}=lim\frac{\sqrt{4+\frac{1}{n}+\frac{1}{n^2}}-1}{n+\frac{2}{n}}=\frac{1}{\infty}=0\)
\(lim\frac{\sqrt{9n^2+n+1}-2n}{3n^2+2}=\frac{\sqrt{9+\frac{1}{n}+\frac{1}{n^2}}-2}{3n+\frac{2}{n}}=\frac{1}{\infty}=0\)
NV
Nguyễn Việt Lâm
Giáo viên
18 tháng 2 2020
Muốn giúp bạn lắm mà ko sao dịch được đề :D
Bạn sử dụng công cụ gõ công thức, nó ở ngoài cùng bên trái khung soạn thảo, chỗ khoanh đỏ ấy, cực dễ sử dụng
Y
15 tháng 10 2023
\(1,\lim\limits_{n\rightarrow\infty}\dfrac{-n^2+2n+1}{\sqrt{3n^4+2}}\left(1\right)\)
\(\dfrac{-n^2+2n+1}{\sqrt{3n^4+2}}=\dfrac{-\dfrac{n^2}{n^4}+\dfrac{2n}{n^4}+\dfrac{1}{n^4}}{\sqrt{\dfrac{3n^4}{n^4}+\dfrac{2}{n^4}}}=\dfrac{-\dfrac{1}{n^2}+\dfrac{2}{n^3}+\dfrac{1}{n^4}}{\sqrt{3+\dfrac{2}{n^4}}}\)
\(\Rightarrow\left(1\right)=\dfrac{-lim\dfrac{1}{n^2}+2lim\dfrac{1}{n^3}+lim\dfrac{1}{n^4}}{\sqrt{lim\left(3+\dfrac{2}{n^4}\right)}}\)
\(=\dfrac{0}{\sqrt{lim\left(3+\dfrac{2}{n^4}\right)}}=0\)
\(2,\lim\limits_{n\rightarrow\infty}\left(\dfrac{4n-\sqrt{16n^2+1}}{n+1}\right)\left(2\right)\)
\(\dfrac{4n-\sqrt{16n^2+1}}{n+1}=\dfrac{\dfrac{4n}{n^2}-\sqrt{\dfrac{16n^2}{n^2}+\dfrac{1}{n^2}}}{\dfrac{n}{n^2}+\dfrac{1}{n^2}}=\dfrac{\dfrac{4}{n}-\sqrt{16+\dfrac{1}{n^2}}}{\dfrac{1}{n}+\dfrac{1}{n^2}}\)
\(\Rightarrow\left(2\right)=\dfrac{lim\left(\dfrac{4}{n}-\sqrt{16+\dfrac{1}{n^2}}\right)}{lim\left(\dfrac{1}{n}+\dfrac{1}{n^2}\right)}=\dfrac{lim\left(\dfrac{4}{n}-\sqrt{16+\dfrac{1}{n^2}}\right)}{0}\)
Vậy giới hạn \(\left(2\right)\) không xác định.
\(3,\lim\limits_{n\rightarrow\infty}\left(\dfrac{\sqrt{9n^2+n+1}-3n}{2n}\right)\left(3\right)\)
\(\dfrac{\sqrt{9n^2+n+1}-3n}{2n}=\dfrac{\sqrt{9+\dfrac{1}{n}+\dfrac{1}{n^2}}-\dfrac{3}{n}}{\dfrac{2}{n}}\)
\(\Rightarrow\left(3\right)=\dfrac{lim\left(\sqrt{9+\dfrac{1}{n}+\dfrac{1}{n^2}}-\dfrac{3}{n}\right)}{2lim\dfrac{1}{n}}=\dfrac{lim\left(\sqrt{9+\dfrac{1}{n}+\dfrac{1}{n^2}}-\dfrac{3}{n}\right)}{0}\)
Vậy \(lim\left(3\right)\) không xác định.
=3/4
3/4 nha bạn