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\(=\lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+2x}-\sqrt{x^2+x}+x-\sqrt{x^2+x}\right)\)
\(=\lim\limits_{x\rightarrow+\infty}\left(\dfrac{x}{\sqrt{x^2+2x}+\sqrt{x^2+x}}-\dfrac{x}{x+\sqrt{x^2+x}}\right)\)
\(=\lim\limits_{x\rightarrow+\infty}\left(\dfrac{1}{\sqrt{1+\dfrac{2}{x}}+\sqrt{1+\dfrac{1}{x}}}-\dfrac{1}{1+\sqrt{1+\dfrac{1}{x}}}\right)=\dfrac{1}{2}-\dfrac{1}{2}=0\)
\(=\lim\limits_{x\rightarrow+\infty}\frac{x+\sqrt{x+\sqrt{x}}-x}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}\)
\(=\lim\limits_{x\rightarrow+\infty}\frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}\)
\(=\lim\limits_{x\rightarrow+\infty}\frac{\sqrt{1+\sqrt{\frac{1}{x}}}}{\sqrt{1+\sqrt{\frac{1}{x}+\sqrt{\frac{1}{x^3}}}}+1}=\frac{1}{1+1}=\frac{1}{2}\)
Giới hạn này tiến đến đâu vậy bạn? 2 trường hợp khác nhau đúng ko?
\(\lim\limits_{x\rightarrow+\infty}\frac{\sqrt{x^2+3x+5}}{\sqrt[3]{x^3+7x^2+8}}=\lim\limits_{x\rightarrow+\infty}\frac{x\sqrt{1+\frac{3}{x}+\frac{5}{x^2}}}{x\sqrt[3]{1+\frac{7}{x}+\frac{8}{x^3}}}=1\)
\(\lim\limits_{x\rightarrow-\infty}\frac{\sqrt{x^2+3x+5}}{\sqrt[3]{x^3+7x^2+8}}=\lim\limits_{x\rightarrow-\infty}\frac{\left|x\right|\sqrt{1+\frac{3}{x}+\frac{5}{x^2}}}{x\sqrt[3]{1+\frac{7}{x}+\frac{8}{x^3}}}=\lim\limits_{x\rightarrow-\infty}\frac{-x\sqrt{1+\frac{3}{x}+\frac{5}{x^2}}}{x\sqrt[3]{1+\frac{7}{x}+\frac{8}{x^3}}}=-1\)
Hai trường hợp sẽ cho ra 2 kết quả khác nhau bạn
\(\lim\limits_{x\rightarrow-\infty}\left(\sqrt{x^2+1}+x-1\right)\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{x^2+1-\left(x-1\right)^2}{\sqrt{x^2+1}-x+1}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{x^2+1-x^2+2x-1}{-x\sqrt{1+\dfrac{1}{x^2}}-x+1}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{-2x}{x\left(-\sqrt{1+\dfrac{1}{x^2}}-1+\dfrac{1}{x}\right)}\)
\(=\lim\limits_{x\rightarrow-\infty}\dfrac{-2}{-\sqrt{1+\dfrac{1}{x^2}}-1+\dfrac{1}{x}}\)
\(=\dfrac{-2}{-\sqrt{1+0}-1+0}=\dfrac{-2}{-1-1}=1\)
b: \(\lim\limits\dfrac{\sqrt{4n^2+n-1}+n}{\sqrt{n^4+2n^3-1}-n}\)
\(=\lim\limits\dfrac{n\left(\sqrt{4+\dfrac{1}{n}-\dfrac{1}{n^2}}+1\right)}{n^2\cdot\sqrt{1+\dfrac{2}{n}-\dfrac{1}{n^4}}-n^2\cdot\dfrac{1}{n}}\)
\(=\lim\limits\dfrac{n\left(\sqrt{4+\dfrac{1}{n}-\dfrac{1}{n^2}}+1\right)}{n^2\left(\sqrt{1+\dfrac{2}{n}-\dfrac{1}{n^4}}-\dfrac{1}{n}\right)}\)
\(=\lim\limits\dfrac{\sqrt{4+\dfrac{1}{n}-\dfrac{1}{n^2}}+1}{n\left(\sqrt{1+\dfrac{2}{n}-\dfrac{1}{n^4}}-\dfrac{1}{n}\right)}\)
\(=\lim\limits\dfrac{\sqrt{\dfrac{4}{n^2}+\dfrac{1}{n^3}-\dfrac{1}{n^4}}+\dfrac{1}{n}}{\sqrt{1+\dfrac{2}{n}-\dfrac{1}{n^4}}-\dfrac{1}{n}}\)
\(=\dfrac{0}{\sqrt{1+0-0}-0}=\dfrac{0}{1}=0\)
Bạn tự hiểu là giới hạn khi x tiến tới dương vô cực
\(=lim\left[x\left(\sqrt{1-\frac{3}{x}+\frac{5}{x^2}}+a\right)\right]=lim\left[x\left(1-a\right)\right]\)
Do \(x\rightarrow+\infty\) nên để giới hạn đã cho bằng \(+\infty\Leftrightarrow1-a>0\Rightarrow a< 1\)
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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\(\lim\limits_{x\to 2019}\frac{\sqrt{x+285}-48}{\sqrt{x-2018}-\sqrt{2020-x}}=\lim\limits_{x\to -\infty}(\sqrt{x+285}-48).\lim\limits_{x\to -\infty}\frac{1}{\sqrt{x-2018}-\sqrt{2020-x}}\)
\(=\lim\limits_{x\to 2019}\frac{x-2019}{\sqrt{x+285}+48}.\lim\limits_{x\to 2019}\frac{\sqrt{x-2018}+\sqrt{2020-x}}{2(x-2019)}=\lim\limits_{x\to 2019}\frac{\sqrt{x-2018}+\sqrt{2020-x}}{2(\sqrt{x+285}+48)}=\frac{1}{96}\)