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\(\lim\limits_{x\rightarrow3}\frac{2\left(\sqrt{x+1}-2\right)}{x-3}=\lim\limits_{x\rightarrow3}\frac{2\left(\sqrt{x+1}-2\right)\left(\sqrt{x+1}+2\right)}{\left(x-3\right)\left(\sqrt{x+1}+2\right)}=\lim\limits_{x\rightarrow3}\frac{2\left(x-3\right)}{\left(x-3\right)\left(\sqrt{x+1}+2\right)}\)
\(=\lim\limits_{x\rightarrow3}\frac{2}{\sqrt{x+1}+2}=\frac{2}{4}=\frac{1}{2}\)
Nếu
\(\lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+x+1}-2\sqrt{x^2-x+1}\right)=\lim\limits_{x\rightarrow+\infty}x\left(\sqrt{1+\frac{1}{x}+\frac{1}{x^2}}-2\sqrt{1-\frac{1}{x}+\frac{1}{x^2}}\right)\)
\(=+\infty.\left(1-2\right)=-\infty\)
Nếu:
\(\lim\limits_{x\rightarrow-\infty}\left(\sqrt{x^2+x+1}-2\sqrt{x^2-x+1}\right)=\lim\limits_{x\rightarrow-\infty}x\left(-\sqrt{1+\frac{1}{x}+\frac{1}{x^2}}+2\sqrt{1-\frac{1}{x}+\frac{1}{x^2}}\right)\)
\(=-\infty.\left(-1+2\right)=-\infty\)
\(\lim\limits_{x\rightarrow+\infty}\frac{\sqrt{x^2+1}+x}{3x+5}=\lim\limits_{x\rightarrow+\infty}\frac{\sqrt{1+\frac{1}{x^2}}+1}{3+\frac{5}{x}}=\frac{2}{3}\)
\(\lim\limits_{x\rightarrow2}\frac{\left(x-2\right)\left(x+2\right)}{x-2}=\lim\limits_{x\rightarrow2}\left(x+2\right)=4\)
16.
\(y'=\frac{\left(cos2x\right)'}{2\sqrt{cos2x}}=\frac{-2sin2x}{2\sqrt{cos2x}}=-\frac{sin2x}{\sqrt{cos2x}}\)
17.
\(y'=4x^3-\frac{1}{x^2}-\frac{1}{2\sqrt{x}}\)
18.
\(y'=3x^2-2x\)
\(y'\left(-2\right)=16;y\left(-2\right)=-12\)
Pttt: \(y=16\left(x+2\right)-12\Leftrightarrow y=16x+20\)
19.
\(y'=-\frac{1}{x^2}=-x^{-2}\)
\(y''=2x^{-3}=\frac{2}{x^3}\)
20.
\(\left(cotx\right)'=-\frac{1}{sin^2x}\)
21.
\(y'=1+\frac{4}{x^2}=\frac{x^2+4}{x^2}\)
22.
\(lim\left(3^n\right)=+\infty\)
11.
\(\lim\limits_{x\rightarrow1^+}\frac{-2x+1}{x-1}=\frac{-1}{0}=-\infty\)
12.
\(y=cotx\Rightarrow y'=-\frac{1}{sin^2x}\)
13.
\(y'=2020\left(x^3-2x^2\right)^{2019}.\left(x^3-2x^2\right)'=2020\left(x^3-2x^2\right)^{2019}\left(3x^2-4x\right)\)
14.
\(y'=\frac{\left(4x^2+3x+1\right)'}{2\sqrt{4x^2+3x+1}}=\frac{8x+3}{2\sqrt{4x^2+3x+1}}\)
15.
\(y'=4\left(x-5\right)^3\)
\(B=\lim\limits_{x\rightarrow+\infty}x\left(\frac{\left(\sqrt{x^2+2x}+x\right)^2-4\left(x^2+x\right)}{\sqrt{x^2+2x}+x+2\sqrt{x^2+x}}\right)\)
\(=\lim\limits_{x\rightarrow+\infty}2x^2\left(\frac{\sqrt{x^2+2x}-x-1}{\sqrt{x^2+2x}+x+2\sqrt{x^2+x}}\right)\)
\(=\lim\limits_{x\rightarrow+\infty}\frac{2x^2\left(x^2+2x-\left(x+1\right)^2\right)}{\left(\sqrt{x^2+2x}+x+2\sqrt{x^2+x}\right)\left(\sqrt{x^2+2x}+x+1\right)}\)
\(=\lim\limits_{x\rightarrow+\infty}\frac{-2x^2}{\left(\sqrt{x^2+2x}+x+2\sqrt{x^2+x}\right)\left(\sqrt{x^2+2x}+x+1\right)}\)
\(=\lim\limits_{x\rightarrow+\infty}\frac{-2x^2}{x^2\left(\sqrt{1+\frac{2}{x}}+1+2\sqrt{1+\frac{1}{x}}\right)\left(\sqrt{1+\frac{2}{x}}+1+\frac{1}{x}\right)}=\frac{-2}{\left(1+1+2\right)\left(1+1+0\right)}=-\frac{1}{4}\)
Đáp án C