Bài 2 trang 79

Đề bài

Tính các giới hạn sau:

a) lim2n2+6n+18n2+5$lim\frac{2n^2+6n+1}{8n^2+5}$                            

b) lim4n2−3n+1−3n3+6n2−2$lim\frac{4n^2-3n+1}{-3n^3+6n^2-2}$;          

c) lim√4n2−n+38n−5$lim\frac{\sqrt{4n^2-n+3}}{8n-5}$;

d) lim(4−2n+13n)$lim\left(4-\frac{2^{n+1}}{3^n}\right)$                           

e) lim4.5n+2n+26.5n$lim\frac{{4.5}^n+2^{n+2}}{{6.5}^n}$                         

g) lim2+4n36n$lim\frac{2+\frac{4}{n^3}}{6^n}$.

Lời giải chi tiết

a) lim2n2+6n+18n2+5=limn2(2+6n+1n2)n2(8+5n2)=lim2+6n+1n8+5n=28=14$lim\frac{2n^2+6n+1}{8n^2+5}=lim\frac{n^2\left(2+\frac{6}{n}+\frac{1}{n^2}\right)}{n^2\left(8+\frac{5}{n^2}\right)}=lim\frac{2+\frac{6}{n}+\frac{1}{n}}{8+\frac{5}{n}}=\frac{2}{8}=\frac{1}{4}$

b) lim4n2−3n+1−3n3+6n2−2=limn3(4n−3n2+1n3)n3(−3+6n−2n3)=lim4n−3n2+1n3−3+6n−2n3=0−0+0−3+0−0=0$lim\frac{4n^2-3n+1}{-3n^3+6n^2-2}=lim\frac{n^3\left(\frac{4}{n}-\frac{3}{n^2}+\frac{1}{n^3}\right)}{n^3\left(-3+\frac{6}{n}-\frac{2}{n^3}\right)}=lim\frac{\frac{4}{n}-\frac{3}{n^2}+\frac{1}{n^3}}{-3+\frac{6}{n}-\frac{2}{n^3}}=\frac{0-0+0}{-3+0-0}=0$.

c) lim√4n2−n+38n−5=limn√4−1n+3n2n(8−5n)=√4−0+08−0=28=14$lim\frac{\sqrt{4n^2-n+3}}{8n-5}=lim\frac{n\sqrt{4-\frac{1}{n}+\frac{3}{n^2}}}{n\left(8-\frac{5}{n}\right)}=\frac{\sqrt{4-0+0}}{8-0}=\frac{2}{8}=\frac{1}{4}$.

d) lim(4−2n+13n)=lim(4−2⋅(23)n)=4−2.0=4$lim\left(4-\frac{2^{n+1}}{3^n}\right)=lim\left(4-2\cdot {\left(\frac{2}{3}\right)}^n\right)=4-2.0=4$.

e) lim4.5n+2n+26.5n=lim4.5n+22.2n6.5n=lim5n.[4+4.(25)n]6.5n=lim4+4.(25)n6=4+4.06=23$lim\frac{{4.5}^n+2^{n+2}}{{6.5}^n}=lim\frac{{4.5}^n+2^2{.2}^n}{{6.5}^n}=lim\frac{5^n.\left[4+4.{\left(\frac{2}{5}\right)}^n\right]}{{6.5}^n}=lim\frac{4+4.{\left(\frac{2}{5}\right)}^n}{6}=\frac{4+4.0}{6}=\frac{2}{3}$.

g) lim2+4n36n=lim(2+4n3).lim(16)n=(2+0).0=0$lim\frac{2+\frac{4}{n^3}}{6^n}=lim\left(2+\frac{4}{n^3}\right).lim{\left(\frac{1}{6}\right)}^n=\left(2+0\right).0=0$.