21.
\(\lim\limits_{x\rightarrow-1}\dfrac{x^2+1}{x-1}=\dfrac{\left(-1\right)^2+1}{-1-1}=-1\)
\(lim\dfrac{2^n-4\cdot3^n}{2\cdot3^n-1}=lim\dfrac{\left(\dfrac{2}{3}\right)^n-4}{2-\dfrac{1}{3^n}}=-\dfrac{4}{2}=-2\)
=> đáp án A
23.
\(\lim\limits_{x\rightarrow6}\dfrac{\sqrt{x-2}-2}{6-x}=\lim\limits_{x\rightarrow6}\dfrac{x-6}{\left(6-x\right)\left(\sqrt{x-2}+2\right)}=\lim\limits_{x\rightarrow6}\dfrac{1}{\sqrt{x-2}+2}=\dfrac{1}{\sqrt{6-2}+2}=\dfrac{1}{4}\)
17.
\(\lim\limits\sqrt{x}=\lim\limits x^{\dfrac{1}{2}}=\dfrac{1}{2}x^{-\dfrac{1}{2}}=\dfrac{1}{2}.\dfrac{1}{x^{\dfrac{1}{2}}}=\dfrac{1}{2\sqrt{x}}\)
16.
\(lim\dfrac{1+3+5+...+\left(2n-1\right)}{2n^2-1}=lim\dfrac{n^2}{2n^2-1}=lim\dfrac{1}{2-\dfrac{1}{n^2}}=\dfrac{1}{2-0}=\dfrac{1}{2}\)
15.
\(\lim\limits_{x\rightarrow+\infty}x^4=+\infty\)
14.
\(f'\left(x\right)=\left(x^3+x^2+x+12\right)'\)
\(=\left(x^3\right)'+\left(x^2\right)'+\left(x\right)'+\left(12\right)'\)
\(=3x^2+2x+1\)
\(f'\left(x\right)=3x^2+2x+1< 6\)
\(\Leftrightarrow\left(x-1\right)\left(3x+5\right)< 0\)
\(\Leftrightarrow-\dfrac{5}{3}< x< 1\)
12.
\(lim\dfrac{2^n-4.3^n}{2.3^n-1}=lim\dfrac{\left(\dfrac{2}{3}\right)^n-4}{2-\dfrac{1}{3^n}}=\dfrac{0-4}{2-0}=-2\)
19.
\(f\left(x\right)=3x^2-1\)
\(\lim\limits_{x\rightarrow1}f\left(x\right)=\lim\limits_{x\rightarrow1}\left(3x^2-1\right)=3.1^2-1=2\)
20.
Câu này cũng không biết như nào, phân vân giữa B và C.
B thì chắc chắn sai, \(\lim\limits q^n=0\left(q< 1\right)\) mới đúng.
C thì nếu \(n=1\) thì \(\lim\limits\dfrac{1}{n^k}=\lim\limits\dfrac{1}{1^k}=\lim\limits1=1\).
22.
Hệ số góc: \(y'=-3x^2+1\Rightarrow y'\left(-2\right)=-11\)
