Giải bài 20 trang 76 sách bài tập toán 11 - Cánh diều

Đề bài

Tính các giới hạn sau:

a) $\underset{x\to -1}{lim}\left(-4x^2+3x+1\right)$                   

b) $\underset{x\to -1}{lim}\frac{-4x+1}{x^2-x+3}$

 c) $\underset{x\to 2}{lim}\sqrt{3x^2+5x+4}$                                   

d) $\underset{x\to -\mathrm{\infty }}{lim}\frac{-3+\frac{4}{x}}{2x^2+3}$

e) $\underset{x\to 2^{+}}{lim}\frac{-3}{x-2}$                                         

g) $\underset{x\to {\left(-2\right)}^{+}}{lim}\frac{5}{x+2}$

Lời giải chi tiết

a) Ta có $\underset{x\to -1}{lim}\left(-4x^2+3x+1\right)=\underset{x\to -1}{lim}\left(-4x^2\right)+\underset{x\to -1}{lim}3x+\underset{x\to -1}{lim}1=-4+\left(-3\right)+1=-6$

b) Ta có $\underset{x\to -1}{lim}\frac{-4x+1}{x^2-x+3}=\frac{\underset{x\to -1}{lim}\left(-4x\right)+\underset{x\to -1}{lim}1}{\underset{x\to -1}{lim}{x}^2+\underset{x\to -1}{lim}\left(-x\right)+\underset{x\to -1}{lim}3}=\frac{4+1}{1+1+3}=1$

c) Xét $\underset{x\to 2}{lim}\left(3x^2+5x+4\right)=\underset{x\to 2}{lim}3x^2+\underset{x\to 2}{lim}5x+\underset{x\to 2}{lim}4={3.2}^2+5.2+4=26$

Suy ra $\underset{x\to 2}{lim}\sqrt{3x^2+5x+4}=\sqrt{26}$.

d) Ta có:$\underset{x\to -\mathrm{\infty }}{lim}\frac{-3+\frac{4}{x}}{2x^2+3}=\underset{x\to -\mathrm{\infty }}{lim}\frac{x^2\left(\frac{-3}{x^2}+\frac{4}{x^3}\right)}{x^2\left(2+\frac{3}{x^2}\right)}=\underset{x\to -\mathrm{\infty }}{lim}\frac{\frac{-3}{x^2}+\frac{4}{x^3}}{2+\frac{3}{x^2}}$

$=\frac{\underset{x\to -\mathrm{\infty }}{lim}\frac{-3}{x^2}+\underset{x\to -\mathrm{\infty }}{lim}\frac{4}{x^3}}{\underset{x\to -\mathrm{\infty }}{lim}2+\underset{x\to -\mathrm{\infty }}{lim}\frac{3}{x^2}}=\frac{0+0}{2+0}=0$

e) Ta có $\underset{x\to 2^{+}}{lim}\frac{-3}{x-2}=-\mathrm{\infty }$

f) Ta có $\underset{x\to {\left(-2\right)}^{+}}{lim}\frac{5}{x+2}=-\mathrm{\infty }$