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a/ \(=\lim\limits_{x\rightarrow\pm\infty}\dfrac{\dfrac{\left(2x\right)^2.\left(4x\right)^3}{x^4}}{\dfrac{\left(3x\right)^2\left(5x^2\right)}{x^4}}=\lim\limits_{x\rightarrow\pm\infty}\dfrac{4^4.x}{45}=\pm\infty\)
b/ \(=\lim\limits_{x\rightarrow\pm\infty}\dfrac{\sqrt[3]{\dfrac{x^3}{x^3}+\dfrac{2x^2}{x^3}+\dfrac{x}{x^3}}}{\dfrac{2x}{x}-\dfrac{2}{x}}=\dfrac{1}{2}\)
c/ \(=\lim\limits_{x\rightarrow\pm\infty}\dfrac{\dfrac{\sqrt[3]{\left(x^3+2x^2\right)^2}}{x^2}+\dfrac{x\sqrt[3]{x^3+2x^2}}{x^2}+\dfrac{x^2}{x^2}}{\dfrac{3x^2}{x^2}-\dfrac{2x}{x^2}}=\dfrac{1+1+1}{3}=1\)
d/ \(=\lim\limits_{x\rightarrow+\infty}\dfrac{\dfrac{\left(-3x\right)^3x^2}{x^5}}{-\dfrac{4x^5}{x^5}}=\dfrac{-27}{-4}=\dfrac{27}{4}\)
e/ \(=\lim\limits_{x\rightarrow+\infty}\dfrac{\dfrac{\left(2x\right)^{20}.\left(3x\right)^{20}}{x^{50}}}{\dfrac{\left(2x\right)^{50}}{x^{50}}}=0\)
g/ \(=\lim\limits_{x\rightarrow+\infty}\dfrac{\dfrac{8x^3.\left(4x^5\right)^9}{x^{47}}}{\dfrac{11x^{47}}{x^{47}}}=+\infty\)
Lời giải:
\(\lim\limits_{x\to \pm\infty}\sqrt{x^2-3x+4}=\lim\limits_{x\to \pm\infty}\sqrt{x^2}.\lim\limits_{x\to \pm \infty}\sqrt{1-\frac{3}{x}+\frac{4}{x^2}}=\lim\limits_{x\to \pm\infty}|x|.1=+\infty \)
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\(\lim\limits_{x\to +\infty}x(\sqrt{x^2+5}+x)=\lim\limits_{x\to +\infty}x^2.\lim\limits_{x\to +\infty}(\sqrt{1+\frac{5}{x^2}}+1)=2(+\infty )=+\infty \)
\(\lim\limits_{x\to -\infty}x(\sqrt{x^2+5}+x)=\lim\limits_{x\to -\infty}\frac{5x}{\sqrt{x^2+5}-x}=\lim\limits_{x\to -\infty}\frac{-5}{\sqrt{1+\frac{5}{x^2}}+1}=\frac{-5}{2}\)
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\(\lim\limits_{x\to 2019}\frac{\sqrt{x+285}-48}{\sqrt{x-2018}-\sqrt{2020-x}}=\lim\limits_{x\to -\infty}(\sqrt{x+285}-48).\lim\limits_{x\to -\infty}\frac{1}{\sqrt{x-2018}-\sqrt{2020-x}}\)
\(=\lim\limits_{x\to 2019}\frac{x-2019}{\sqrt{x+285}+48}.\lim\limits_{x\to 2019}\frac{\sqrt{x-2018}+\sqrt{2020-x}}{2(x-2019)}=\lim\limits_{x\to 2019}\frac{\sqrt{x-2018}+\sqrt{2020-x}}{2(\sqrt{x+285}+48)}=\frac{1}{96}\)
Lời giải:
\(\lim\limits_{x\to \pm\infty}\sqrt{x^2-3x+4}=\lim\limits_{x\to \pm\infty}\sqrt{x^2}.\lim\limits_{x\to \pm \infty}\sqrt{1-\frac{3}{x}+\frac{4}{x^2}}=\lim\limits_{x\to \pm\infty}|x|.1=+\infty \)
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\(\lim\limits_{x\to +\infty}x(\sqrt{x^2+5}+x)=\lim\limits_{x\to +\infty}x^2.\lim\limits_{x\to +\infty}(\sqrt{1+\frac{5}{x^2}}+1)=2(+\infty )=+\infty \)
\(\lim\limits_{x\to -\infty}x(\sqrt{x^2+5}+x)=\lim\limits_{x\to -\infty}\frac{5x}{\sqrt{x^2+5}-x}=\lim\limits_{x\to -\infty}\frac{-5}{\sqrt{1+\frac{5}{x^2}}+1}=\frac{-5}{2}\)
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\(\lim\limits_{x\to 2019}\frac{\sqrt{x+285}-48}{\sqrt{x-2018}-\sqrt{2020-x}}=\lim\limits_{x\to -\infty}(\sqrt{x+285}-48).\lim\limits_{x\to -\infty}\frac{1}{\sqrt{x-2018}-\sqrt{2020-x}}\)
\(=\lim\limits_{x\to 2019}\frac{x-2019}{\sqrt{x+285}+48}.\lim\limits_{x\to 2019}\frac{\sqrt{x-2018}+\sqrt{2020-x}}{2(x-2019)}=\lim\limits_{x\to 2019}\frac{\sqrt{x-2018}+\sqrt{2020-x}}{2(\sqrt{x+285}+48)}=\frac{1}{96}\)
\(2cos2x+2cosx-\sqrt{2}=0\\ \Leftrightarrow2\left(2cos^2x-1\right)+2cosx-\sqrt{2}=0\\ \Leftrightarrow4cos^2x+2cosx-2-\sqrt{2}=0\\ \Leftrightarrow\left[{}\begin{matrix}cosx=\frac{\sqrt{2}}{2}\\cosx=-\frac{1+\sqrt{2}}{2}\left(l\right)\end{matrix}\right.\\ \Rightarrow x=\pm\frac{\pi}{4}+k2\pi,k\in Z\)
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Th1: \(\lim\limits_{x\rightarrow+\infty}\dfrac{x}{\sqrt{x^2+x+1}}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{1}{\sqrt{1+\dfrac{1}{x}+\dfrac{1}{x^2}}}\)
\(=\dfrac{1}{\sqrt{1+0+0}}=\dfrac{1}{1}=1\)
TH2: \(\lim\limits_{x\rightarrow-\infty}\dfrac{x}{\sqrt{x^2+x+1}}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{x}{-x\sqrt{1+\dfrac{1}{x}+\dfrac{1}{x^2}}}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{-1}{\sqrt{1+\dfrac{1}{x}+\dfrac{1}{x^2}}}=\dfrac{-1}{1}=-1\)
TH1: \(\lim\limits_{x\rightarrow+\infty}\dfrac{x+1}{\sqrt{x^2-x+1}}\)
\(=\lim\limits_{x\rightarrow+\infty}\dfrac{1+\dfrac{1}{x}}{\sqrt{1-\dfrac{1}{x}+\dfrac{1}{x^2}}}\)
\(=\dfrac{1+0}{\sqrt{1-0+0}}=\dfrac{1}{1}=1\)
TH2: \(\lim\limits_{x\rightarrow-\infty}\dfrac{x+1}{\sqrt{x^2-x+1}}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{x+1}{-x\cdot\sqrt{1-\dfrac{1}{x}+\dfrac{1}{x^2}}}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{1+\dfrac{1}{x}}{-\sqrt{1-\dfrac{1}{x}+\dfrac{1}{x^2}}}=\dfrac{1+0}{-\sqrt{1-0+0}}=\dfrac{1}{-1}=-1\)
TL:
Take [math]a*x^2 + b*x +c = 0[/math]
Then
=>[math]x^2 + \frac{b}{a} x + \frac{c}{a} = 0[/math]
=>[math]x^2 + \frac{2b}{2a} x + \frac{c}{a} = 0[/math]
=>[math]x^2 + \frac{2b}{2a} x + (\frac{b}{2a})^2 + \frac{c}{a} - ( (\frac{b}{2a})^2 = 0[/math] -(1)
We have it in the form of [math]x^2 + 2px + p^2 + q = 0[/math]
which is [math](x+p)^2 + q = 0[/math]