I.

Do \(\left(u_n\right)\) là cấp số nhân \(\Rightarrow\)\(u_4=u_3.q\Rightarrow q=\dfrac{u_4}{u_3}=\dfrac{10}{3}\)

\(u_3=u_1q^2\Rightarrow u_1=\dfrac{u_3}{q^2}=\dfrac{27}{100}\)

2. Công thức số hạng tổng quát: \(u_n=\dfrac{27}{100}.\left(\dfrac{10}{3}\right)^{n-1}\)

II.

1. \(\lim\limits\dfrac{-3n^2+2n-2022}{3n^2-2022}=\lim\dfrac{-3+\dfrac{2}{n}-\dfrac{2022}{n^2}}{3-\dfrac{2022}{n^2}}=\dfrac{-3+0-0}{3-0}=-1\)

2.

\(\lim\limits_{x\rightarrow2}\dfrac{x^2-5x+6}{x-2}=\lim\limits_{x\rightarrow2}\dfrac{\left(x-2\right)\left(x-3\right)}{x-2}=\lim\limits_{x\rightarrow2}\left(x-3\right)=-1\)

\(=\lim\limits_{x\rightarrow2}x-1=2-1=1\)

\(\lim\limits_{x\rightarrow a}\dfrac{x^4-a^4}{x^2-a^2}=\lim\limits_{x\rightarrow a}\left(x^2+a^2\right)=2a^2\)

\(\left(...\right)=\lim\limits_{x\rightarrow1}\dfrac{2\left(x-1\right)}{\left(x-1\right)\left(\sqrt{2x+7}+3\right)}=\lim\limits_{x\rightarrow1}\dfrac{2}{\sqrt{2x+7}+3}=\dfrac{1}{3}\)

1: \(-1< =cosx< =1\)

=>\(-3< =3\cdot cosx< =3\)

=>\(y\in\left[-3;3\right]\)

2:

TXĐ là D=R

3: \(L=\lim\limits\dfrac{-3n^3+n^2}{2n^3+5n-2}\)

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

4:

\(L=lim\left(3n^2+5n-3\right)\)

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

\(=+\infty\) vì \(\left\{{}\begin{matrix}lim\left(n^2\right)=+\infty\\\lim\limits\left(3+\dfrac{5}{n}-\dfrac{3}{n^2}\right)=3>0\end{matrix}\right.\)

5:

\(\lim\limits_{n\rightarrow+\infty}n^3-2n^2+3n-4\)

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

\(=+\infty\) vì \(\left\{{}\begin{matrix}\lim\limits_{n\rightarrow+\infty}n^3=+\infty\\\lim\limits_{n\rightarrow+\infty}1-\dfrac{2}{n}+\dfrac{3}{n^2}-\dfrac{4}{n^3}=1>0\end{matrix}\right.\)

\(1,y=3cosx\)

\(+TXD\) \(D=R\)

Có \(-1\le cosx\le1\)

\(\Leftrightarrow-3\le3cosx\le3\)

Vậy có tập giá trị \(T=\left[-3;3\right]\)

\(2,y=cosx\)

\(TXD\) \(D=R\)

\(3,L=lim\dfrac{n^2-3n^3}{2n^3+5n-2}=lim\dfrac{\dfrac{1}{n}-3}{2+\dfrac{5}{n^2}-\dfrac{2}{n^3}}\)(chia cả tử và mẫu cho \(n^3\))

\(=\dfrac{lim\dfrac{1}{n}-lim3}{lim2+5lim\dfrac{1}{n^2}-2lim\dfrac{1}{n^3}}=\dfrac{0-3}{2+5.0-2.0}=-\dfrac{3}{2}\)

\(4,L=lim\left(3n^2+5n-3\right)\\ =lim\left(3+\dfrac{5}{n}-\dfrac{3}{n^2}\right)\\ =lim3+5lim\dfrac{1}{n}-3lim\dfrac{1}{n^2}\\ =3\)

\(5,\lim\limits_{n\rightarrow+\infty}\left(n^3-2n^2+3n-4\right)\\ =lim\left(1-\dfrac{2}{n}+\dfrac{3}{n^2}-\dfrac{4}{n^3}\right)\\ =lim1-0\\ =1\)

1: \(\lim\limits_{x\rightarrow4}\dfrac{1-x}{\left(x-4\right)^2}=-\infty\) 

vì \(\left\{{}\begin{matrix}\lim\limits_{x\rightarrow4}1-x=1-4=-3< 0\\\lim\limits_{x\rightarrow4}\left(x-4\right)^2=\left(4-4\right)^2=0\end{matrix}\right.\)

2: \(\lim\limits_{x\rightarrow3^+}\dfrac{2x-1}{x-3}=+\infty\)

vì \(\left\{{}\begin{matrix}\lim\limits_{x\rightarrow3^+}2x-1=2\cdot3-1=5>0\\\lim\limits_{x\rightarrow3^+}x-3=3-3>0\end{matrix}\right.\) và x-3>0

3: \(\lim\limits_{x\rightarrow2^+}\dfrac{-2x+1}{x+2}\)

\(=\dfrac{-2\cdot2+1}{2+2}=\dfrac{-3}{4}\)

4: \(\lim\limits_{x\rightarrow1^-}\dfrac{3x-1}{x+1}=\dfrac{3\cdot1-1}{1+1}=\dfrac{2}{2}=1\)

1.

\(\lim\limits_{x\rightarrow-1}\dfrac{2x^2-x-3}{x^2-1}=\lim\limits_{x\rightarrow-1}\dfrac{\left(x+1\right)\left(2x-3\right)}{\left(x+1\right)\left(x-1\right)}=\lim\limits_{x\rightarrow-1}\dfrac{2x-3}{x-1}=\dfrac{5}{2}\)

2.

a. \(y'=6x^2-sinx-\dfrac{1}{2\sqrt{x}}\)

b. \(y'=10\left(x^2-5\right)^9.\left(x^2-5\right)'=20x\left(x^2-5\right)^9\)

3.

\(y'=-2x\)

\(k=4\Rightarrow-2x=4\Rightarrow x=-2\Rightarrow y\left(-2\right)=-24\)

Phương trình tiếp tuyến:

\(y=4\left(x+2\right)-24\Leftrightarrow y=4x-16\)

Khi x → +∞

Khi x → -∞