Bài 1: Giới hạn của dãy số

Cho dãy số \(a_n=\sqrt[3]{n^3+1}-n\). Hãy tính \(\lim\limits a_n\).

1. \(0\).
2. \(\dfrac{1}{3}\).
3. \(\dfrac{1}{2}\).
4. \(1\).

Cho dãy số \(a_n=\sqrt[3]{n^3-3n^2+1}-\sqrt{n^2+4n}\). Hãy tính \(\lim\limits a_n\) .

1. \(2\).
2. \(-2\).
3. \(3\).
4. \(-3\).

Cho dãy số \(a_n=\dfrac{4.3^n+7^{n+1}}{2.5^n+7^n}\). Tính \(\lim\limits a_n\).

1. \(\dfrac{11}{3}\).
2. \(7\).
3. \(\dfrac{5}{3}\).
4. \(2\).

Tính  \(\lim\limits\dfrac{1+a+a^2+....+a^n}{1+b+b^2+...+b^n}\) biết rằng \(\left|a\right|< 1;\left|b\right|< 1\).

1. \(\dfrac{b}{a}\).
2. \(\dfrac{1+b}{1+a}\).
3. \(\dfrac{1-b}{1-a}\).
4. \(\dfrac{1-a}{1-b}\).

Tính \(\lim\limits\left(\dfrac{1+3+5+...+2n-1}{n+1}-\dfrac{2n+1}{2}\right)\).

1. \(\dfrac{3}{2}\).
2. \(1\).
3. \(-1\).
4. \(-\dfrac{3}{2}\).

Tính \(\lim\limits a_n\) biết dãy số \(a_n=\dfrac{n.\sin n!}{n^2+1}\) .

1. \(\lim\limits a_n=1\).
2. \(\lim\limits a_n=\dfrac{1}{2}\).
3. \(\lim\limits a_n=+\infty\).
4. \(\lim\limits a_n=0\).

\(lim\dfrac{n^2-2\sqrt{n+1}cosn}{n^2+2n}\)  là​

1. 1.
2. 2.
3. \(\dfrac{1}{3}\).
4. \(\dfrac{1}{4}\).

\(lim\left(n^4-2n.sin3n+1\right)\) là​

1. \(1\).
2. \(-\infty\).
3. \(+\infty\).
4. \(10^5\).

\(\lim\dfrac{2^n.3^n-3^n.4^n+5^{2n}}{3^n.4^n-4^n.5^n+6^{2n}}\) là​

1. \(0\).
2. \(-2\).
3. 2.
4. \(\dfrac{1}{2}\).

\(\lim\left(\sqrt{2n^2+2n-1}-\sqrt{2n^2-n-5}\right)\) bằng

1. \(\dfrac{3\sqrt{2}}{4}\).
2. \(\dfrac{3\sqrt{2}}{2}\).
3. \(\dfrac{3\sqrt{2}}{5}\).
4. \(\dfrac{3\sqrt{2}}{10}\).

Tính \(\lim\left(\sqrt{2n^2-n+1}-\sqrt{2n^2-4n+5}\right).n\).​

1. \(+\infty\).
2. \(-\infty\).
3. \(1\).
4. \(2\).

Giới hạn nào trong bốn giới hạn sau đây có giá trị bằng 2 ?​

1. \(lim\left(2n-cosn\right)\).
2. \(lim\dfrac{4n^2-2n.cos3n+2}{\sqrt{4n^4+2n^2+8}}\).
3. \(lim\left(n^2+n-4\sqrt[3]{n}\right)\).
4. \(lim\dfrac{3n^2+n}{n+2}\).

Tính \(\lim\dfrac{1}{\sqrt{4n+2}-\sqrt{2n+1}}\).​

1. 0.
2. 1.
3. 2.
4. \(\dfrac{1}{2}\).

Tính  \(\lim\dfrac{\left(2n+1\right)\left(n^2+3n\right)}{\left(n+1\right)^2.n}\) là​

1. 2.
2. 3.
3. 1.
4. 4.

Tính \(\lim\left(3+\dfrac{\left(-1\right)^n.\sin\left(3n\right)}{n^2+1}\right)\).​

1. 3.
2. \(+\infty\).
3. \(-\infty\).
4. \(0\).​

Tính  \(\lim(\sqrt{3n.3^n-3^n}+8)\) .​

1. 1.
2. 2.
3. 3.
4. \(+\infty\).

\(\lim\dfrac{1}{5n+3}\) bằng​

1. \(\dfrac{1}{5}\).
2. \(+\infty\).
3. \(\dfrac{1}{3}\).
4. \(0\).

Tính \(\lim\limits\dfrac{6n-1}{3n+2}\).

1. 2.
2. -1.
3. \(\dfrac{1}{2}\).
4. -2.

Tính \(\lim\limits\dfrac{3n^4-2n+4}{4n^2+2n+3}\).

1. \(0\).
2. \(+\infty\).
3. \(\dfrac{3}{4}\).
4. \(\dfrac{2}{7}\).

Tính \(\lim\limits\dfrac{\sqrt{2x+3}}{\sqrt{2x}+5}\).

1. \(\dfrac{5}{7}\).
2. \(\dfrac{5}{2}\).
3. 1.
4. \(+\infty\).