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Lời giải:
$\frac{u_{n-1}}{u_n}=\frac{n^2}{n^2-1}>0$ với mọi $n\geq 2$ nên $u_{n-1}, u_n$ luôn cùng dấu.
Mà $u_1=2017>0$ nên $u_n>0$ với mọi $n=1,2,...$
Mặt khác:
$n^2(u_{n-1}-u_n)=u_{n-1}>0\Rightarrow u_{n-1}>u_n$ nên dãy $(u_n)$ là dãy giảm.
Dãy giảm và bị chặn dưới nên $u_n$ hội tụ. Đặt $\lim u_n=a$.
Ta có: $a=n^2(a-a)\Rightarrow a=0$
Vậy $\lim u_n=0$
Ủa đề bài như này là sao bạn? Cho dãy x(k), nhưng lại đi tìm u(n)?
Đề bài có nghĩa là tìm giới hạn dãy:
\(u_n=\sqrt[n]{\left(\dfrac{1}{2!}\right)^n+\left(\dfrac{1}{2!}+\dfrac{2}{3!}\right)^n+...+\left(\dfrac{1}{2!}+\dfrac{2}{3!}+...+\dfrac{2011}{2012!}\right)^n}\)
Tìm được khá dễ dàng bằng cách sử dụng định lý kẹp
\(\lim\limits_{x\rightarrow1}\frac{x^{2016}+x-2}{\sqrt{2018x+1}-\sqrt{x+2018}}=\lim\limits_{x\rightarrow1}\frac{2016x^{2015}+1}{\frac{1009}{\sqrt{2018x+1}}-\frac{1}{2\sqrt{x+2018}}}=\frac{2017}{\frac{1009}{\sqrt{2019}}-\frac{1}{2\sqrt{2019}}}=2\sqrt{2019}\)
Để hàm liên tục tại \(x=1\)
\(\Rightarrow\lim\limits_{x\rightarrow1}f\left(x\right)=f\left(1\right)\Rightarrow k=2\sqrt{2019}\)
2.
\(\lim\limits_{x\rightarrow1}\frac{x^2+ax+b}{x^2-1}=\frac{1}{2}\Leftrightarrow\left\{{}\begin{matrix}a+b+1=0\\\lim\limits_{x\rightarrow1}\frac{2x+a}{2x}=\frac{1}{2}\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a+b=-1\\\frac{a+2}{2}=\frac{1}{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=-1\\b=0\end{matrix}\right.\) \(\Rightarrow S=1\)
3.
\(\lim\limits_{x\rightarrow1}\frac{\sqrt{x^2+x+2}-2+2-\sqrt[3]{7x+1}}{\sqrt{2}\left(x-1\right)}=\lim\limits_{x\rightarrow1}\frac{\frac{\left(x-1\right)\left(x+2\right)}{\sqrt{x^2+x+2}+2}-\frac{7\left(x-1\right)}{\sqrt[3]{\left(7x+1\right)^2}+2\sqrt[3]{7x+1}+4}}{\sqrt{2}\left(x-1\right)}\)
\(=\lim\limits_{x\rightarrow1}\frac{1}{\sqrt{2}}\left(\frac{x+2}{\sqrt{x^2+x+2}+2}-\frac{7}{\sqrt[3]{\left(7x+1\right)^2}+2\sqrt[3]{7x+1}+4}\right)\)
\(=\frac{1}{\sqrt{2}}\left(\frac{3}{4}-\frac{7}{12}\right)=\frac{\sqrt{2}}{12}\)
\(\Rightarrow a+b+c=1+12+0=13\)
Chọn A
Phương pháp: Tìm công thức số hạng tổng quát
Cách giải: Ta có:
u ( 1 ) = 1
u ( 2 ) = u ( 1 ) + u ( 1 ) = 2 u ( 1 ) + 1
u ( 3 ) = u ( 2 ) + u ( 1 ) = 3 u ( 1 ) + 1 + 2
u ( 4 ) = u ( 3 ) + u ( 1 ) = 4 u ( 1 ) + 1 + 2 + 3
. . .
u ( 2017 ) = u ( 2016 ) + u ( 1 ) = 2017 u ( 1 ) + 1 + 2 + 3 . . . + 2016
⇒ u ( 2017 ) = 1 + 2 + 3 . . . + 2016 + 2017 = 2035153
\(a=\lim\limits_{x\rightarrow1}\frac{\left(x-1\right)\left(x+1\right)\left(x^2+1\right)}{\left(x-1\right)\left(x^2+x-1\right)}=\lim\limits_{x\rightarrow1}\frac{\left(x+1\right)\left(x^2+1\right)}{x^2+x-1}=\frac{4}{1}=4\)
\(b=\lim\limits_{x\rightarrow-1}\frac{\left(x+1\right)\left(x^4-x^3+x^2-x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}=\lim\limits_{x\rightarrow-1}\frac{x^4-x^3+x^2-x+1}{x^2-x+1}=\frac{5}{3}\)
\(c=\lim\limits_{x\rightarrow3}\frac{\left(x+1\right)\left(x-3\right)^2}{\left(x^2+1\right)\left(x^2-9\right)}=\lim\limits_{x\rightarrow3}\frac{\left(x+1\right)\left(x-3\right)}{\left(x^2+1\right)\left(x+3\right)}=\frac{0}{60}=0\)
\(d=\lim\limits_{x\rightarrow1}\frac{4x^6-5x^5+x}{x^2-2x+1}=\lim\limits_{x\rightarrow1}\frac{24x^5-25x^4+1}{2x-2}=\lim\limits_{x\rightarrow1}\frac{120x^4-100x^3}{2}=10\)
\(e=\lim\limits_{x\rightarrow1}\frac{mx^{m-1}}{nx^{n-1}}=\frac{m}{n}\)
\(f=\lim\limits_{x\rightarrow-2}\frac{\left(x+2\right)\left(x-2\right)\left(x^2+4\right)}{\left(x+2\right)x^2}=\lim\limits_{x\rightarrow-2}\frac{\left(x-2\right)\left(x^2+4\right)}{x^2}=-8\)
Hai câu d, e khai triển thì dài quá nên làm biếng sử dụng L'Hopital
\(A=\lim\limits_{x\rightarrow2}\frac{\left(x-2\right)\left(2x-1\right)}{x-2}=\lim\limits_{x\rightarrow2}\left(2x-1\right)=3\)
\(B=\lim\limits_{x\rightarrow1}\frac{\left(x-1\right)\left(x^2-2x+3\right)}{\left(x-1\right)\left(x+1\right)}=\lim\limits_{x\rightarrow1}\frac{x^2-2x+3}{x+1}=\frac{1-2+3}{1+1}=1\)
\(C=\lim\limits_{x\rightarrow2}\frac{x^2+2x}{x^2+4x+4}=\frac{4+4}{4+8+4}=\frac{1}{2}\)
\(D=\lim\limits_{x\rightarrow1}\frac{\left(x-1\right)\left(x^2-1\right)}{\left(x-1\right)\left(x-2\right)}=\lim\limits_{x\rightarrow1}\frac{x^2-1}{x-2}=\frac{0}{-1}=0\)
\(E=\lim\limits_{x\rightarrow1}\frac{x^3-5x^2+3x+9}{x^4-8x^4-9}=\frac{1-5+3+9}{1-8-9}=-\frac{1}{2}\)
\(F=\lim\limits_{x\rightarrow-1}\frac{\left(x+1\right)\left(x-1\right)\left(x^2+1\right)}{\left(x+1\right)\left(x^2-3x+3\right)}=\lim\limits_{x\rightarrow-1}\frac{\left(x-1\right)\left(x^2+1\right)}{x^2-3x+3}=\frac{-2.2}{1+3+3}=-\frac{2}{5}\)
\(G=\lim\limits_{x\rightarrow1}\frac{\left(x-1\right)\left(x+3\right)}{\left(x-1\right)\left(2x+1\right)}=\lim\limits_{x\rightarrow1}\frac{x+3}{2x+1}=\frac{4}{3}\)
\(H=\lim\limits_{x\rightarrow-2}\frac{\left(x+2\right)\left(x-1\right)^2}{\left(2-x\right)\left(x+2\right)}=\lim\limits_{x\rightarrow-2}\frac{\left(x-1\right)^2}{2-x}=\frac{9}{4}\)
\(I=\lim\limits_{x\rightarrow1}\frac{4x^6-5x^5+1}{x^2-1}=\lim\limits_{x\rightarrow1}\frac{24x^5-25x^4}{2x}=\frac{24-25}{2}=-\frac{1}{2}\)
\(K=\lim\limits_{x\rightarrow1}\frac{x^m-1}{x^n-1}=\lim\limits_{x\rightarrow1}\frac{mx^{m-1}}{nx^{n-1}}=\frac{m}{n}\)
