6n+9\(⋮\)4n-1 ->4.(6n+9)\(⋮\)4n-1

->24n+36\(⋮\)4n-1

->24n-6+42\(⋮\)4n-1

->6(4n-1)+42\(⋮\)4n-1

->4n-1 thuoc uoc cua 42 ma n\(\supseteq\)1 nen 4n-1\(\supseteq\)3

ma n laf so tu nhien nen n=1,2

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

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

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

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

a)4n²-3n-1 chia hết cho 4n-1

<=>4n²-n-2n-1 chia hết cho 4n-1

<=>n(4n-1)-(2n+1) chia hết cho 4n-1

<=>2n+1 chia hết cho 4n-1

<=>2(2n+1) chia hết cho 4n-1

<=>4n-1+3 chia hết cho 4n-1

<=>3 chia hết cho 4n-1

=>4n-1 thuộc Ư(3)

=>Ư(3)={-1;1;-3;3}

Ta có bảng sau:

Vậy n thuộc {0;1}

b)4n²-3n-1 chia hết cho n-1

<=>4n²-4n+n-1 chia hết cho n-1

<=>4n(n-1)+n-1 chia hết cho n-1

<=>(4n+1)(n-1) chia hết cho n-1

<=>n thuộc N với mọi gtrị

P/s: "chia hết cho" thì viết kí hiệu vô

Is that T :))

a.\(2n^2-3n+1=2n\times\left(n-1\right)-\left(n-1\right)=\left(2n-1\right)\times\left(n-1\right)\Rightarrow2n-1⋮n-1\)

\(\Rightarrow2\left(n-1\right)+1⋮n-1\Rightarrow1⋮n-1\Rightarrow n-1\inƯ\left(1\right)=\left\{1\right\}\Rightarrow n=2\)

b.Tách tương tự nha

\(2n^2-3n+1=\left(2n^2-2n\right)-n+1=2n\left(n-1\right)-n+1\)\(\Rightarrow-n+1⋮n-1\Rightarrow-\left(n-1\right)⋮n-1\)

vậy với mọi x thuộc N đều t/m

b) tương tự nha

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

\(=\lim\dfrac{\left(2+\dfrac{1}{n}\right)\left(3-\dfrac{2}{n}\right)^2}{1+\dfrac{1}{n^2}-\dfrac{1}{n^3}}=\dfrac{2.3^2}{1}=18\)

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

a)

\(n+5⋮n+1\)

\(\Rightarrow n+1+4⋮n+1\)

\(\Rightarrow4⋮n+1\Rightarrow n+1\inƯ\left(4\right)=\left\{\pm1;\pm2;\pm4\right\}\)

\(\Rightarrow n\in\left\{0;-2;1;-3;3;-5\right\}\)

\(a,\left(n+5\right)⋮\left(n+1\right)\Leftrightarrow\left(n+1\right)+4⋮\left(n+1\right)\)

\(\Leftrightarrow4⋮n+1\left(n\inℤ\right)\)

\(\Leftrightarrow n+1\inƯ\left(4\right)=\left\{\pm1;\pm2;\pm4\right\}\)

\(\Leftrightarrow n=-2;0;-3;1;-5;3\)

Vậy \(n=-5;-3;-2;0;1;3\)

a) Ta có: \(\frac{n+5}{n+1}=\frac{n+1+4}{n+1}=1+\frac{4}{n+1}\)

Để (n+5) chia hết cho (n+1)

Thì 4 phải chia hết cho n+1

\(\Rightarrow n+1\inƯ\left(4\right)=\left\{\pm1;\pm2;\pm4\right\}\)

Lập bảng ta có:

Vậy số nguyên n thỏa mãn là

n = {-5;-3;-2;0;1;3}

b) \(\frac{6n+4}{2n+1}=\frac{3\left(2n+1\right)+1}{2n+1}=3+\frac{1}{2n+1}\)

Để (6n+4) chia hết cho (2n+1)

Thì 1 phải chia hết cho 2n+1

\(\Rightarrow2n+1\inƯ\left(1\right)=\left\{1;-1\right\}\)

Vậy n = {-1;0}

\(A=\dfrac{6n+3-2}{2n+1}=3-\dfrac{2}{2n+1}\)

Để A max thì 2/2n+1 min

mà n nguyên

nên 2n+1=-1

=>2n=-2

=>n=-1