Bài tập 3 trang 84 SBT Toán 11 Tập 1 Chân trời sáng tạo - CTST

a) Ta có: $\underset{x\to -2}{\lim }\frac{x^2-4}{x+2}$

$=\underset{x\to -2}{\lim }\frac{\left(x+2\right)\left(x-2\right)}{x+2}$

$=\underset{x\to -2}{\lim }\left(x-2\right)$

$=-2-2$

$=-4.$

b) Ta có: $\underset{x\to 1}{\lim }\frac{x^3-1}{1-x}$

$=\underset{x\to 1}{\lim }\frac{\left(x-1\right)\left(x^2+x+1\right)}{-\left(x-1\right)}$

$=\underset{x\to 1}{\lim }\left[-\left(x^2+x+1\right)\right]$

$=-\left(\underset{x\to 1}{\lim }{x}^2+\underset{x\to 1}{\lim }x+1\right)$

$\left(\mathrm{}\right)=-3.$

c) Ta có: Ta có: $\underset{x\to 3}{\lim }\frac{x^2-4x+3}{x-3}$

$=\underset{x\to 3}{\lim }\frac{\left(x-1\right)\left(x-3\right)}{x-3}$

$=\underset{x\to 3}{\lim }\left(x-1\right)$

$=3-1$

$=2$

d) Ta có: $\underset{x\to -2}{\text{lim}}\frac{2-\sqrt{x+6}}{x+2}$

$=\underset{x\to -2}{\text{lim}}\frac{\left(2-\sqrt{x+6}\right)\left(2+\sqrt{x+6}\right)}{\left(x+2\right)\left(2+\sqrt{x+6}\right)}$

$=\underset{x\to -2}{\text{lim}}\frac{4-\left(x+6\right)}{\left(x+2\right)\left(2+\sqrt{x+6}\right)}$

$=\underset{x\to -2}{\text{lim}}\frac{-\left(x+2\right)}{\left(x+2\right)\left(2+\sqrt{x+6}\right)}$

$=\underset{x\to -2}{\text{lim}}\frac{-1}{2+\sqrt{x+6}}$

$=\frac{-1}{2+\sqrt{-2+6}}$

$=-\frac{1}{4}.$

e) Ta có: $\underset{x\to 0}{\text{lim}}\frac{x}{\sqrt{x+1}-1}$

$=\underset{x\to 0}{\text{lim}}\frac{x\left(\sqrt{x+1}+1\right)}{\left(\sqrt{x+1}-1\right)\left(\sqrt{x+1}+1\right)}$

$=\underset{x\to 0}{\text{lim}}\frac{x\left(\sqrt{x+1}+1\right)}{\left(x+1\right)-1}$

$=\underset{x\to 0}{\text{lim}}\left(\sqrt{x+1}+1\right)$

$=2.$

g) Ta có: $\underset{x\to 2}{\lim }\frac{x^2-4x+4}{x^2-4}$

$=\underset{x\to 2}{\lim }\frac{{\left(x-2\right)}^2}{\left(x+2\right)\left(x-2\right)}$

$=\underset{x\to 2}{\lim }\frac{x-2}{x+2}$

$=\frac{\underset{x\to 2}{\lim }\left(x-2\right)}{\underset{x\to 2}{\lim }\left(x+2\right)}$

$=\frac{0}{4}=0.$

-- Mod Toán 11 HỌC247