99+ bài tập giới hạn của dãy số Toán 11

Bài 1. Tính các giới hạn sau:

1. $ \displaystyle \lim \frac{2n^3-n^2+1}{n^3+1} $
2. $ \displaystyle \lim \frac{-n^7-n^6+1}{n+2n^7} $
3. $ \displaystyle \lim \frac{(n+1)(n^2-3n+5)}{n^3-2n(n^2+1)+2} $
4. $\displaystyle \lim \frac{n\sqrt{n}+n^2-1}{2n^2+1} $
5. $\displaystyle \lim \frac{n+\sqrt{n^2+1}}{3n-1} $
6. $\displaystyle \lim \frac{n+\sqrt{n^2+2n+1}}{2-\sqrt{4n^2+1}} $
7. $\displaystyle \lim \frac{2n+\sqrt{n^3+2n^2+1}}{n\sqrt{3n+2}-1}$
8. $\displaystyle \lim \frac{\sqrt[3]{-27n^6+2n+1}}{4n^2+4n+1}$
9. $\displaystyle \lim \sqrt{\frac{3n^2+2n-1}{n^2+5n}} $
10. $\displaystyle \lim \left(\frac{2n^2}{n^2+3n+1}-\frac{2n}{3n+1}\right) $
11. $\displaystyle \lim \frac{n+1}{n^3+1} $
12. $ \displaystyle \lim \frac{11n^2-2n+1}{n^3+n^2+1} $
13. $\displaystyle \lim \frac{(2n+1)(n-5)+n^2+1}{n^3+n^2} $
14. $ \displaystyle \lim \left(\frac{2n}{3n^2+1}-\frac{n}{3n^2+1}\right) $
15. $\displaystyle \lim \frac{n+\sqrt{n^3+1}}{2n^2+\sqrt{n}-1} $
16. $\displaystyle \lim \frac{2}{n(\sqrt{n^2+1}-n)} $
17. $\displaystyle \lim \frac{1}{\sqrt{n}\left(\sqrt{n+2}-\sqrt{n+1}\right)} $
18. $\displaystyle \lim \frac{3}{\sqrt{4n^2+1}-2n+1} $
19. $\displaystyle \lim \frac{2^n+3^n}{5\cdot3^n+2^n} $
20. $\displaystyle \lim \frac{2^n-1}{3^n+2^{n+1}} $
21. $\displaystyle \lim \frac{2^n-3^n+5^{n+2}}{5^n+3^{n+1}} $
22. $ \displaystyle \lim \frac{(-2)^n-5^{n+1}}{5^{n-1}+3^{n+1}}$

Bài 2. Tính các giới hạn sau:

1. $\lim \dfrac{n^2+2n}{3n^2+n+1}$
2. $\lim \dfrac{n\sqrt[3]{n^3+2}-4}{5n^2+1}$
3. $\lim \dfrac{(2n\sqrt{n}+1)(\sqrt{n}+3)}{(n+1)(n-2)}$
4. $\lim \dfrac{(n+1)(3n^2+2n-1)}{4n^3-5n}$
5. $\lim \dfrac{(2n-1)(3n+2)(4n-3)}{5n^4+7}$
6. $\lim \dfrac{\sqrt[4]{n^4+3n-1}+2n}{4n+3}$
7. $\lim \dfrac{\sqrt[3]{n^3+2n^2}}{2n+1}$
8. $\lim \dfrac{\sqrt{n^2+3}-n-4}{\sqrt{n^2+2}+n}$
9. $\lim \dfrac{(\sqrt{n^2+1}+n)^2}{\sqrt[3]{n^6+n}}$
10. $\lim \dfrac{2n^2+1}{n^3-3n-3}$
11. $\lim \dfrac{2n-1}{(n^2+1)(3n+2)}$
12. $\lim \dfrac{3n^2-5}{(n-3)(2n^2+3)}$
13. $\lim \dfrac{{\sqrt[3]{{{n^3} + {n^2} + n}} + 3\sqrt {{n^2} + 1} }}{{n\sqrt 3 + 1}}$
14. $ \lim (n^2-3n+1) $
15. $ \lim (-3n^2-3n+1) $
16. $ \lim (n^2-3n^3+n) $
17. $ \lim (-2n^2-3n+\frac{1}{n}) $
18. $ \lim (n\sqrt{n}-3n^2+1) $
19. $ \lim (\sqrt{n^2+1}+2n)$

Bài 3. Tính các giới hạn sau:

1. $\lim(\sqrt{n^2+n}-n)$
2. $\lim(\sqrt{n^2+2n}-n+1)$
3. $\lim(\sqrt{n^2+3n}-n-2)$
4. $\lim(\sqrt{4n^2+3n+1}-2n+1)$
5. $\lim(\sqrt{9n^2+5n}-3n+1)$
6. $\lim(\sqrt[3]{n^3+2n^2-n}-n)$
7. $\lim(\sqrt[3]{8n^3+3n^2+1}+1-2n)$
8. $\lim(\sqrt[3]{2n-n^3}+n-1)$
9. $\lim(1+n^2-\sqrt{n^4+3n+1})$
10. $\lim n(\sqrt[3]{n^3+5n}-n)$
11. $ \lim \sqrt {n – 1} (\sqrt {n + 1} – \sqrt n ) $
12. $\lim n(\sqrt{n^2+1}-\sqrt{n^2-4})$

Bài 4. Tính các giới hạn sau:

1. $ \lim \dfrac{2^n-5.3^n}{3^n+1}$
2. $ \lim \dfrac{2^n+2^{n+1}}{2^n+4.3^n} $
3. $ \lim \dfrac{4.3^n+7^{n+1}}{2.5^n+7^n} $
4. $\lim \dfrac{{{3^n} – {4^{n + 1}}}}{{{3^{n + 2}} + {4^n}}}$
5. $ \lim \dfrac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+2}} $
6. $\lim \dfrac{{{2^n} + {6^n} – {4^{n + 1}}}}{{{3^n} + {6^{n + 1}}}}$
7. $\lim \dfrac{{{3^n} – {4^n} + {5^n}}}{{{3^n} + {4^n} – {5^n}}}$
8. $\lim \dfrac{{ – 3{n^2} + 4n + 1}}{{{n^2}\cdot{2^n}}}$

Bài 5. Tính các giới hạn sau:

1. $\lim\dfrac{1+2+3+…+n}{n}$
2. $\lim \dfrac{{1 + 3 + 5 + … + (2n + 1)}}{{3{n^2} + 4}}$
3. $\lim \dfrac{{1 + 2 + 3 + … + n}}{{{n^2} – 3}}$
4. $\lim \dfrac{{{1^2} + {2^2} + {3^2} + … + {n^2}}}{{n(n + 1)(n + 2)}}$
5. $\lim \left({\dfrac{1}{{1.2}} + \dfrac{1}{{2.3}} +\cdots+ \dfrac{1}{{n(n + 1)}}} \right)$
6. $\lim(\sqrt{n^2+3n}-n-2)$
7. $\lim(\sqrt{4n^2+3n+1}-2n+1)$
8. $\lim(\sqrt{9n^2+5n}-3n+1)$
9. $\lim(\sqrt[3]{n^3+2n^2-n}-n)$
10. $\lim(\sqrt[3]{8n^3+3n^2+1}+1-2n)$
11. $\lim(\sqrt[3]{2n-n^3}+\sqrt{n^2-1})$
12. $ \lim \left(\sqrt{n^2+n}-\sqrt[3]{n^3+2n}\right)$
13. $ \lim \left(\sqrt{n^2+n}+\sqrt[3]{1-n^3}\right)$$\lim(1+n^2-\sqrt{n^4+3n+1})$
14. $\lim n(\sqrt[3]{n^3+5n}-n)$
15. $\lim n(\sqrt{n^2+1}-\sqrt{n^2-4}$