Câu 1.
lim \(\varkappa\rightarrow-\infty\) (\(\sqrt{\varkappa^2-7\varkappa+1}-\sqrt{\varkappa^2-3\varkappa+2}\))=?
Câu 2.
lim \(\frac{\varkappa+\sqrt{\varkappa}}{\varkappa-\sqrt{\varkappa}}\)=?
\(\varkappa\rightarrow0^+\)
Câu 3.
lim \(\frac{\varkappa^2-3}{\varkappa^3+2}\)=?
\(\varkappa\rightarrow-1\)
Câu 4.
tìm m để hs f(x)\(\left\{{}\begin{matrix}\frac{2\varkappa^2-\varkappa-1}{\varkappa-1}\\m,\end{matrix}\right.\)khi\(\varkappa\ne1,m=1\)
có giới hạn khi x tiến dần về 1?
Câu 5.
lim \(\frac{-3\varkappa^5+7\varkappa^3-11}{\varkappa^5+\varkappa^4-3\varkappa}=?\)
\(\varkappa\rightarrow-\infty\)
Câu 6.
lim \(\frac{\varkappa^2+3\varkappa-4}{\varkappa^2+4\varkappa}=?\)
\(\varkappa\rightarrow-4\)
Câu 7.
lim \(\frac{\varkappa+2}{\varkappa-2}=?\)
\(\varkappa\rightarrow2^-\)
Câu 8.
lim \(\frac{3-\sqrt{2\varkappa+7}}{\varkappa^2-1}=?\)
\(\varkappa\rightarrow1\)
Câu 9.
lim \(\frac{64-\varkappa^3}{4-\varkappa}=?\)
\(\varkappa\rightarrow4\)
1.
\(\lim\limits_{x\rightarrow-\infty}\frac{x^2-7x+1-\left(x^2-3x+2\right)}{\sqrt{x^2-7x+1}+\sqrt{x^2-3x+2}}=\lim\limits_{x\rightarrow-\infty}\frac{-4x-1}{\sqrt{x^2-7x+1}+\sqrt{x^2-3x+2}}\)
\(=\lim\limits_{x\rightarrow-\infty}\frac{x\left(-4-\frac{1}{x}\right)}{-x\sqrt{1-\frac{7}{x}+\frac{1}{x^2}}-x\sqrt{1-\frac{3}{x}+\frac{2}{x^2}}}=\frac{-4}{-1-1}=2\)
2.
\(\lim\limits_{x\rightarrow0^+}\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}=\lim\limits_{x\rightarrow0^+}\frac{\sqrt{x}+1}{\sqrt{x}-1}=-1\)
3.
\(\lim\limits_{x\rightarrow-1}\frac{x^2-3}{x^3+2}=\frac{1-3}{-1+2}=-2\) (ko phải dạng vô định, cứ thay số tính)
4.
\(\lim\limits_{x\rightarrow1}f\left(x\right)=\lim\limits_{x\rightarrow1}\frac{2x^2-x-1}{x-1}=\lim\limits_{x\rightarrow1}\frac{\left(x-1\right)\left(2x+1\right)}{x-1}=\lim\limits_{x\rightarrow1}\left(2x+1\right)=3\)
Để hs có giới hạn tại \(x=1\Rightarrow m=3\)
5.
\(\lim\limits_{x\rightarrow-\infty}\frac{-3x^5+7x^3-11}{x^5+x^4-3x}=\lim\limits_{x\rightarrow-\infty}\frac{-3+\frac{7}{x^2}-\frac{11}{x^5}}{1+\frac{1}{x}-\frac{3}{x^4}}=\frac{-3}{1}=-3\)
6.
\(\lim\limits_{x\rightarrow-4}\frac{\left(x+4\right)\left(x-1\right)}{x\left(x+4\right)}=\lim\limits_{x\rightarrow-4}\frac{x-1}{x}=\frac{-5}{-4}=\frac{5}{4}\)
7.
Khi \(x< 2\Rightarrow x-2< 0\) mà \(x+2\rightarrow4\Rightarrow\lim\limits_{x\rightarrow2^-}\frac{x+2}{x-2}=\frac{4}{-0}=-\infty\)
8.
\(\lim\limits_{x\rightarrow1}\frac{9-\left(2x+7\right)}{\left(x-1\right)\left(x+1\right)\left(3+\sqrt{2x+7}\right)}=\lim\limits_{x\rightarrow1}\frac{-2\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(3+\sqrt{2x+7}\right)}\)
\(=\lim\limits_{x\rightarrow1}\frac{-2}{\left(x+1\right)\left(3+\sqrt{2x+7}\right)}=\frac{-2}{2.\left(3+3\right)}=-\frac{1}{6}\)
9.
\(\lim\limits_{x\rightarrow4}\frac{\left(4-x\right)\left(16-4x+x^2\right)}{4-x}=\lim\limits_{x\rightarrow4}\left(16-4x+x^2\right)=16\)