Sequence Limits — Practice Problems and Worked Solutions

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16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the limit of the sequence.

   $$a_n=\frac{3n+1}{n+2}$$

   Answer: $3$

   Step-by-step solution

   1. $$\text{Compare dominant terms or rationalize the expression.}$$
   2. $$\underset{n\to \infty }{\lim }\frac{3n+1}{n+2}=3$$
2. 2.

   Find the limit of the sequence.

   $$a_n=\frac{2n^2-1}{5n^2+3}$$

   Answer: $\frac{2}{5}$

   Step-by-step solution

   1. $$\text{Compare dominant terms or rationalize the expression.}$$
   2. $$\underset{n\to \infty }{\lim }\frac{2n^2-1}{5n^2+3}=\frac{2}{5}$$
3. 3.

   Find the limit of the sequence.

   $$a_n=\frac{1}{n}$$

   Answer: $0$

   Step-by-step solution

   1. $$\text{Compare dominant terms or rationalize the expression.}$$
   2. $$\underset{n\to \infty }{\lim }\frac{1}{n}=0$$
4. 4.

   Find the limit of the sequence.

   $$a_n=\frac{n+1}{n}$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{Compare dominant terms or rationalize the expression.}$$
   2. $$\underset{n\to \infty }{\lim }\frac{n+1}{n}=1$$
5. 5.

   Find the limit of the sequence.

   $$a_n=\frac{5n^3+n}{2n^3-4}$$

   Answer: $\frac{5}{2}$

   Step-by-step solution

   1. $$\text{Compare dominant terms or rationalize the expression.}$$
   2. $$\underset{n\to \infty }{\lim }\frac{5n^3+n}{2n^3-4}=\frac{5}{2}$$
6. 6.

   Find the limit of the sequence.

   $$a_n=-n+\sqrt{n^2+n}$$

   Answer: $\frac{1}{2}$

   Step-by-step solution

   1. $$\text{Compare dominant terms or rationalize the expression.}$$
   2. $$\underset{n\to \infty }{\lim }-n+\sqrt{n^2+n}=\frac{1}{2}$$
7. 7.

   Find the limit of the sequence.

   $$a_n=\frac{4n+3}{n+1}$$

   Answer: $4$

   Step-by-step solution

   1. $$\text{Compare dominant terms or rationalize the expression.}$$
   2. $$\underset{n\to \infty }{\lim }\frac{4n+3}{n+1}=4$$
8. 8.

   Find the limit of the sequence.

   $$a_n=\frac{3n^2+2}{6n^2-1}$$

   Answer: $\frac{1}{2}$

   Step-by-step solution

   1. $$\text{Compare dominant terms or rationalize the expression.}$$
   2. $$\underset{n\to \infty }{\lim }\frac{3n^2+2}{6n^2-1}=\frac{1}{2}$$
9. 9.

   Find the limit of the sequence.

   $$a_n=\frac{2}{n}$$

   Answer: $0$

   Step-by-step solution

   1. $$\text{Compare dominant terms or rationalize the expression.}$$
   2. $$\underset{n\to \infty }{\lim }\frac{2}{n}=0$$
10. 10.

    Find the limit of the sequence.

    $$a_n=\frac{n+5}{n}$$

    Answer: $1$

    Step-by-step solution

    1. $$\text{Compare dominant terms or rationalize the expression.}$$
    2. $$\underset{n\to \infty }{\lim }\frac{n+5}{n}=1$$
11. 11.

    Find the limit of the sequence.

    $$a_n=\frac{7n^3-n}{2n^3+5}$$

    Answer: $\frac{7}{2}$

    Step-by-step solution

    1. $$\text{Compare dominant terms or rationalize the expression.}$$
    2. $$\underset{n\to \infty }{\lim }\frac{7n^3-n}{2n^3+5}=\frac{7}{2}$$
12. 12.

    Find the limit of the sequence.

    $$a_n=\frac{n^2+1}{n^2+n}$$

    Answer: $1$

    Step-by-step solution

    1. $$\text{Compare dominant terms or rationalize the expression.}$$
    2. $$\underset{n\to \infty }{\lim }\frac{n^2+1}{n^2+n}=1$$
13. 13.

    Find the limit of the sequence.

    $$a_n=\frac{5n-2}{3n+7}$$

    Answer: $\frac{5}{3}$

    Step-by-step solution

    1. $$\text{Compare dominant terms or rationalize the expression.}$$
    2. $$\underset{n\to \infty }{\lim }\frac{5n-2}{3n+7}=\frac{5}{3}$$
14. 14.

    Find the limit of the sequence.

    $$a_n=\frac{1}{n^2}$$

    Answer: $0$

    Step-by-step solution

    1. $$\text{Compare dominant terms or rationalize the expression.}$$
    2. $$\underset{n\to \infty }{\lim }\frac{1}{n^2}=0$$
15. 15.

    Find the limit of the sequence.

    $$a_n=\frac{2n+1}{n^2+3}$$

    Answer: $0$

    Step-by-step solution

    1. $$\text{Compare dominant terms or rationalize the expression.}$$
    2. $$\underset{n\to \infty }{\lim }\frac{2n+1}{n^2+3}=0$$
16. 16.

    Find the limit of the sequence.

    $$a_n=\frac{6n^4+n}{3n^4-2}$$

    Answer: $2$

    Step-by-step solution

    1. $$\text{Compare dominant terms or rationalize the expression.}$$
    2. $$\underset{n\to \infty }{\lim }\frac{6n^4+n}{3n^4-2}=2$$

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