Bài tập 2 trang 79 sgk Toán 11 tập 1 Cánh diều: Tính các giới hạn sau:
a) $\lim \frac{2n^{2}+6n+1}{8n^{2}+5}$;
b) $\lim \frac{4n^{2}-3n+1}{-3n^{3}+6n^{2}-2}$;
c) $\lim \frac{\sqrt{4n^{2}-n+3}}{8n-5}$;
d) $\lim (4-\frac{2^{n+1}}{3^{n}})$;
e) $\lim \frac{4.5^{n}+2^{n+2}}{6.5^{n}}$;
g) $\lim \frac{2+\frac{4}{n^{3}}}{6^{n}}$.
Bài Làm:
a) $\lim \frac{2n^{2}+6n+1}{8n^{2}+5}=\lim \frac{2+\frac{6}{n}+\frac{1}{n^{2}}}{8+\frac{5}{n^{2}}}=\frac{1}{4}$;
b) $\lim \frac{4n^{2}-3n+1}{-3n^{3}+6n^{2}-2}=\lim \frac{n^{2}(4-\frac{3}{n}+\frac{1}{n^{2}})}{n^{3}(-3+\frac{6}{n}-\frac{2}{n^{3}})}=\lim\frac{1}{n}.(-\frac{4}{3})=0$;
c) $\lim \frac{\sqrt{4n^{2}-n+3}}{8n-5}=\lim\frac{\sqrt{4-\frac{1}{n}+\frac{3}{n^{2}}}}{8-\frac{5}{n}}=\frac{1}{4}$;
d) $\lim (4-\frac{2^{n+1}}{3^{n}})=\lim(4-(\frac{2}{3})^{n}.2)=4$;
e) $\lim \frac{4.5^{n}+2^{n+2}}{6.5^{n}}=\lim (\frac{2}{3}+\frac{2}{3}.(\frac{2}{5})^{n})=\frac{2}{3}$;
g) $\lim \frac{2+\frac{4}{n^{3}}}{6^{n}}=\frac{2+0}{+\infty}=0$.
