xin mọi người giúp em vói ạ
xin mọi người giúp em vói ạ
13.
\(\lim\limits_{x\rightarrow1}\left(3x^2-2x-1\right)=3.1^2-2.1-1=0\)
14.
\(\lim\limits_{x\rightarrow9}\sqrt{x+16}=\sqrt{9+16}=5\)
15.
\(\lim\limits_{x\rightarrow+\infty}x^{2021}=+\infty\)
16.
\(\lim\limits_{x\rightarrow-\infty}\left(x-x^3+1\right)=\lim\limits_{x\rightarrow-\infty}x^3\left(-1+\dfrac{1}{x^2}+\dfrac{1}{x^3}\right)\)
Do \(\lim\limits_{x\rightarrow-\infty}x^3=-\infty\)
\(\lim\limits_{x\rightarrow-\infty}\left(-1+\dfrac{1}{x^2}+\dfrac{1}{x^3}\right)=-1< 0\)
\(\Rightarrow\lim\limits_{x\rightarrow-\infty}x^3\left(-1+\dfrac{1}{x^2}+\dfrac{1}{x^3}\right)=+\infty\)
17.
\(\lim\limits_{x\rightarrow-\infty}\dfrac{2x^2+5x-3}{x^2+6x+3}=\lim\limits_{x\rightarrow-\infty}\dfrac{x^2\left(2+\dfrac{5}{x}-\dfrac{3}{x^2}\right)}{x^2\left(1+\dfrac{6}{x}+\dfrac{3}{x^2}\right)}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{2+\dfrac{5}{x}-\dfrac{3}{x^2}}{1+\dfrac{6}{x}+\dfrac{3}{x^2}}=\dfrac{2+0-0}{1+0+0}=2\)
18.
\(\lim\limits_{x\rightarrow-\infty}\dfrac{\sqrt{4x^2-x-1}}{x+1}=\lim\limits_{x\rightarrow-\infty}\dfrac{\left|x\right|\sqrt{4-\dfrac{1}{x}-\dfrac{1}{x^2}}}{x+1}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{-x\sqrt{4-\dfrac{1}{x}-\dfrac{1}{x^2}}}{x\left(1+\dfrac{1}{x}\right)}=\lim\limits_{x\rightarrow-\infty}\dfrac{-\sqrt{4-\dfrac{1}{x}-\dfrac{1}{x^2}}}{1+\dfrac{1}{x}}\)
\(=\dfrac{-\sqrt{4}}{1}=-2\)
19.
\(\lim\limits_{x\rightarrow-\infty}\left(2x^3-x^2\right)=\lim\limits_{x\rightarrow-\infty}x^3\left(2-\dfrac{1}{x}\right)\)
Do \(\lim\limits_{x\rightarrow-\infty}x^3=-\infty\)
\(\lim\limits_{x\rightarrow-\infty}\left(2-\dfrac{1}{x}\right)=2>0\)
\(\Rightarrow\lim\limits_{x\rightarrow-\infty}x^3\left(2-\dfrac{1}{x}\right)=-\infty\)
20.
\(\lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+1}-x\right)=\lim\limits_{x\rightarrow+\infty}\dfrac{\left(\sqrt{x^2+1}-x\right)\left(\sqrt{x^2+1}+x\right)}{\sqrt{x^2+1}+x}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{1}{\sqrt{x^2+1}+x}=\lim\limits_{x\rightarrow+\infty}\dfrac{x\left(\dfrac{1}{x}\right)}{x\left(\sqrt{1+\dfrac{1}{x^2}}+1\right)}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{\dfrac{1}{x}}{\sqrt{1+\dfrac{1}{x^2}}+1}=\dfrac{0}{1+1}=0\)
Với mọi giá trị của tham số m , chứng minh phương trình \(x^5+x^2-\left(m^2+2\right)x-1=0\) luôn có ít nhất 3 nghiệm thực.
Đặt \(f\left(x\right)=x^5+x^2-\left(m^2+2\right)x-1\)
Hàm \(f\left(x\right)\) liên tục trên R
\(f\left(0\right)=-1< 0\)
\(\lim\limits_{x\rightarrow+\infty}f\left(x\right)=\lim\limits_{x\rightarrow+\infty}\left(x^5+x^2-\left(m^2+2\right)x-1\right)\)
\(=\lim\limits_{x\rightarrow+\infty}x^5\left(1+\dfrac{1}{x^3}-\dfrac{m^2+2}{x^4}-\dfrac{1}{x^5}\right)=+\infty.1=+\infty\)
\(\Rightarrow\) Luôn tồn tại \(a>0\) sao cho \(f\left(a\right)>0\Rightarrow f\left(0\right).f\left(a\right)< 0\Rightarrow\) pt luôn có ít nhất 1 nghiệm thuộc \(\left(0;+\infty\right)\)
\(f\left(-1\right)=m^2+1>0;\forall m\Rightarrow f\left(-1\right).f\left(0\right)< 0\Rightarrow\) pt luôn có ít nhất 1 nghiệm thuộc \(\left(-1;0\right)\)
\(\lim\limits_{x\rightarrow-\infty}\left(x^5+x^2-\left(m^2+2\right)x-1\right)=\lim\limits_{x\rightarrow-\infty}x^5\left(1+\dfrac{1}{x^3}-\dfrac{m^2+2}{x^4}-\dfrac{1}{x^5}\right)=-\infty.1=-\infty\)
\(\Rightarrow\) Luôn tồn tại \(b< 0\) sao cho \(f\left(b\right)< 0\Rightarrow f\left(b\right).f\left(-1\right)< 0\Rightarrow\) pt luôn có ít nhất 1 nghiệm thuộc \(\left(-\infty;-1\right)\)
Vậy pt đã cho luôn có ít nhất 3 nghiệm thực
\(\lim\limits_{x\rightarrow3}\dfrac{2-\sqrt[3]{x+5}}{x^2-9}=\lim\limits_{x\rightarrow3}\dfrac{\left(2-\sqrt[3]{x+5}\right)\left(4+2\sqrt[3]{x+5}+\sqrt[3]{\left(x+5\right)^2}\right)}{\left(x-3\right)\left(x+3\right)\left(4+2\sqrt[3]{x+5}+\sqrt[3]{\left(x+5\right)^2}\right)}\)
\(=\lim\limits_{x\rightarrow3}\dfrac{3-x}{\left(x-3\right)\left(x+3\right)\left(4+2\sqrt[3]{x+5}+\sqrt[3]{\left(x+5\right)^2}\right)}\)
\(=\lim\limits_{x\rightarrow3}\dfrac{-1}{\left(x+3\right)\left(4+2\sqrt[3]{x+5}+\sqrt[3]{\left(x+5\right)^2}\right)}\)
\(=\dfrac{-1}{6\left(4+2.2+2^2\right)}=-\dfrac{1}{72}\)
\(\lim\limits_{x\rightarrow1}\dfrac{\sqrt[3]{2x-1}-1}{\sqrt[]{3x+1}-2}=\lim\limits_{x\rightarrow1}\dfrac{\left(\sqrt[3]{2x-1}-1\right)\left(\sqrt[3]{\left(2x-1\right)^2}+\sqrt[3]{2x-1}+1\right)\left(\sqrt[]{3x+1}+2\right)}{\left(\sqrt[]{3x+1}-2\right)\left(\sqrt[]{3x+1}+2\right)\left(\sqrt[3]{\left(2x-1\right)^2}+\sqrt[3]{2x-1}+1\right)}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{2\left(x-1\right)\left(\sqrt[]{3x+1}+2\right)}{3\left(x-1\right)\left(\sqrt[3]{\left(2x-1\right)^2}+\sqrt[3]{2x-1}+1\right)}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{2\left(\sqrt[]{3x+1}+2\right)}{3\left(\sqrt[3]{\left(2x-1\right)^2}+\sqrt[3]{2x-1}+1\right)}\)
\(=\dfrac{2\left(2+2\right)}{3\left(1+1+1\right)}=\dfrac{8}{9}\)
Tính
\(\lim\limits_{x\rightarrow0}\dfrac{\left(x^2+2014\right)\sqrt[2014]{1-2014x}-2014}{x}\)
\(\lim\limits_{x\rightarrow0}\dfrac{\left(x^2+2014\right)\left(1-2014x\right)^{\dfrac{1}{2014}}-2014}{x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{2x\left(1-2014x\right)^{\dfrac{1}{2014}}+\left(x^2+2014\right).\dfrac{1}{2014}.\left(-2014\right).\left(1-2014x\right)^{-\dfrac{2013}{2014}}}{1}\)
\(=-2014\)
Cho dãy un xác định bởi
\(\left\{{}\begin{matrix}x_1=3\\x_{n+1}=\dfrac{1}{2}x_n+2^{n-2}\end{matrix}\right.\) với n= 1,2,3,...
a) Tìm tất cả các số hạng là số nguyên dương trong dãy trên
b) Tìm số hạng tổng quát
\(x_{n+1}=\dfrac{1}{2}x_n+2^{n-2}\Leftrightarrow x_{n+1}-\dfrac{1}{6}.2^{n+1}=\dfrac{1}{2}\left(x_n-\dfrac{1}{6}.2^n\right)\)
Đặt \(x_n-\dfrac{1}{6}.2^n=y_n\Rightarrow\left\{{}\begin{matrix}y_1=x_1-\dfrac{1}{6}.2^1=\dfrac{8}{3}\\y_{n+1}=\dfrac{1}{2}y_n\end{matrix}\right.\)
\(\Rightarrow y_n\) là CSN với công bội \(q=\dfrac{1}{2}\)
\(\Rightarrow y_n=\dfrac{8}{3}.\left(\dfrac{1}{2}\right)^{n-1}=\dfrac{4}{3.2^n}\)
\(\Rightarrow x_n=y_n+\dfrac{1}{6}.2^n=\dfrac{4}{3.2^n}+\dfrac{2^n}{6}\)
Cho dãy un được xác định bởi
\(\left\{{}\begin{matrix}u_1=1\\u_{n+1}=\dfrac{u_n}{u_n+1}\end{matrix}\right.\) với n=1,2,3,.... Tính
\(\lim\limits_{ }\dfrac{2014\left(u_1+1\right)\left(u_2+1\right)....\left(u_n+1\right)}{2015n}\)
\(u_{n+1}=\dfrac{u_n}{u_n+1}\Rightarrow\dfrac{1}{u_{n+1}}=\dfrac{1}{u_n}+1\)
Đặt \(\dfrac{1}{u_n}=v_n\Rightarrow\left\{{}\begin{matrix}v_1=\dfrac{1}{u_1}=1\\v_{n+1}=v_n+1\end{matrix}\right.\)
\(\Rightarrow v_n\) là CSC với công sai \(d=1\Rightarrow v_n=v_1+\left(n-1\right).1=n\)
\(\Rightarrow u_n=\dfrac{1}{n}\)
\(\Rightarrow u_n+1=\dfrac{n+1}{n}\)
\(\lim\dfrac{2014\left(\dfrac{2}{1}\right)\left(\dfrac{3}{2}\right)\left(\dfrac{4}{3}\right)...\left(\dfrac{n+1}{n}\right)}{2015n}=\lim\dfrac{2014\left(n+1\right)}{2015n}=\dfrac{2014}{2015}\)
Mọi người giúp em mấy câu này với ạ!! Một vài câu cũng được mà làm hết thì càng tốt
4.
\(\lim\limits_{x\rightarrow8}f\left(x\right)=\lim\limits_{x\rightarrow8}\dfrac{\sqrt[3]{x}-2}{x-8}=\lim\limits_{x\rightarrow8}\dfrac{x-8}{\left(x-8\right)\left(\sqrt[3]{x^2}+2\sqrt[3]{x}+4\right)}=\lim\limits_{x\rightarrow8}\dfrac{1}{\sqrt[3]{x^2}+2\sqrt[3]{x}+4}\)
\(=\dfrac{1}{4+4+4}=\dfrac{1}{12}\)
\(f\left(8\right)=3.8-20=4\)
\(\Rightarrow\lim\limits_{x\rightarrow8}f\left(x\right)\ne f\left(8\right)\)
\(\Rightarrow\) Hàm gián đoạn tại \(x=8\)
5.
\(\lim\limits_{x\rightarrow0^+}f\left(x\right)=\lim\limits_{x\rightarrow0^+}\dfrac{\sqrt[]{1+2x}-1+1-\sqrt[3]{1+3x}}{x}=\lim\limits_{x\rightarrow0^+}\dfrac{\dfrac{2x}{\sqrt[]{1+2x}+1}-\dfrac{3x}{1+\sqrt[3]{1+3x}+\sqrt[3]{\left(1+3x\right)^2}}}{x}\)
\(=\lim\limits_{x\rightarrow0^+}\left(\dfrac{2}{\sqrt[]{1+2x}+1}-\dfrac{3}{1+\sqrt[3]{1+3x}+\sqrt[3]{\left(1+3x\right)^2}}\right)=\dfrac{2}{1+1}-\dfrac{3}{1+1+1}=0\)
\(f\left(0\right)=\lim\limits_{x\rightarrow0^-}f\left(x\right)=\lim\limits_{x\rightarrow0^-}\left(3x^2-2x\right)=0\)
\(\Rightarrow\lim\limits_{x\rightarrow0^+}f\left(x\right)=\lim\limits_{x\rightarrow0^-}f\left(x\right)=f\left(0\right)\)
\(\Rightarrow\) Hàm liên tục tại \(x=0\)
6.
\(\lim\limits_{x\rightarrow0^+}f\left(x\right)=\lim\limits_{x\rightarrow0^+}\dfrac{\sqrt[]{4x+1}-\sqrt[3]{6x+1}}{x^2}\)
\(=\lim\limits_{x\rightarrow0^+}\dfrac{\sqrt[]{4x+1}-\left(2x+1\right)+\left(2x+1-\sqrt[3]{6x+1}\right)}{x^2}\)
\(=\lim\limits_{x\rightarrow0^+}\dfrac{\dfrac{-x^2}{\sqrt[]{4x+1}+2x+1}+\dfrac{x^2\left(8x+12\right)}{\left(2x+1\right)^2+\left(2x+1\right)\sqrt[3]{6x+1}+\sqrt[3]{\left(6x+1\right)^2}}}{x^2}\)
\(=\lim\limits_{x\rightarrow0^+}\left(\dfrac{-1}{\sqrt[]{4x+1}+2x+1}+\dfrac{8x+12}{\left(2x+1\right)^2+\left(2x+1\right)\sqrt[3]{6x+1}+\sqrt[3]{\left(6x+1\right)^2}}\right)\)
\(=\dfrac{-1}{1+1}+\dfrac{12}{1+1+1}=\dfrac{7}{2}\)
\(f\left(0\right)=\lim\limits_{x\rightarrow0^-}f\left(x\right)=\lim\limits_{x\rightarrow0^-}\left(2-3x\right)=2\)
\(\Rightarrow\lim\limits_{x\rightarrow0^+}f\left(x\right)\ne\lim\limits_{x\rightarrow0^-}f\left(x\right)\)
\(\Rightarrow\) Hàm gián đoạn tại \(x=0\)
7.
\(\lim\limits_{x\rightarrow0^+}f\left(x\right)=\lim\limits_{x\rightarrow0^+}\dfrac{\sqrt[]{1+2x}-\left(x+1\right)+\left(x+1-\sqrt[3]{1+3x}\right)}{x^2}\)
\(=\lim\limits_{x\rightarrow0^+}\dfrac{\dfrac{-x^2}{\sqrt[]{1+2x}+x+1}+\dfrac{x^2\left(x+3\right)}{\left(x+1\right)^2+\left(x+1\right)\sqrt[3]{1+3x}+\sqrt[3]{\left(1+3x\right)^2}}}{x^2}\)
\(=\lim\limits_{x\rightarrow0^+}\left(\dfrac{-1}{\sqrt[]{1+2x}+x+1}+\dfrac{x+3}{\left(x+1\right)^2+\left(x+1\right)\sqrt[3]{1+3x}+\sqrt[3]{\left(1+3x\right)^2}}\right)\)
\(=\dfrac{-1}{1+1}+\dfrac{3}{1+1+1}=1\)
\(f\left(0\right)=\lim\limits_{x\rightarrow0^-}f\left(x\right)=\lim\limits_{x\rightarrow0^-}\left(2x+3\right)=3\)
\(\Rightarrow\lim\limits_{x\rightarrow0^+}f\left(x\right)\ne\lim\limits_{x\rightarrow0^-}f\left(x\right)\)
\(\Rightarrow\) Hàm gián đoạn tại \(x=0\)
chứng minh rằng phương trình (m2+m+4)x2017 -2x+1=0 luôn có ít nhất 1 nghiệm âm với mọi giá trị của tham số m
Đặt \(f\left(x\right)=\left(m^2+m+4\right)x^{2017}-2x+1\)
\(f\left(x\right)\) là hàm đa thức nên liên tục trên R
\(f\left(0\right)=1>0\)
\(m^2+m+4=\left(m+\dfrac{1}{2}\right)^2+\dfrac{15}{4}>0\)
\(\Rightarrow\lim\limits_{x\rightarrow-\infty}\left[\left(m^2+m+4\right)x^{2017}-2x+1\right]=\lim\limits_{x\rightarrow-\infty}x^{2017}\left[\left(m^2+m+4\right)-\dfrac{2}{x^{2016}}+\dfrac{1}{x^{2017}}\right]=-\infty< 0\)
\(\Rightarrow\) Luôn tồn tại 1 số âm \(a< 0\) sao cho \(f\left(a\right)< 0\)
\(\Rightarrow f\left(a\right).f\left(0\right)< 0\)
\(\Rightarrow f\left(x\right)\) luôn có ít nhất 1 nghiệm thuộc \(\left(a;0\right)\)
Hay pt đã cho luôn có ít nhất 1 nghiệm âm với mọi m
Lim x-- dương vô cùng: (4x^2 -3x+1)
\(\lim\limits_{x\rightarrow+\infty}\left(4x^2-3x+1\right)=\lim\limits_{x\rightarrow+\infty}x^2\left(4-\dfrac{3}{x}+\dfrac{1}{x^2}\right)\)
Do \(\left\{{}\begin{matrix}\lim\limits_{x\rightarrow+\infty}x^2=+\infty\\\lim\limits_{x\rightarrow+\infty}\left(4-\dfrac{3}{x}+\dfrac{1}{x^2}\right)=4>0\end{matrix}\right.\)
\(\Rightarrow\lim\limits_{x\rightarrow+\infty}x^2\left(4-\dfrac{3}{x}+\dfrac{1}{x^2}\right)=+\infty\)