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## Answered question

2022-01-06

Determine the limits of the following sequences:
a. ${a}_{n}=\frac{3{n}^{3}}{{n}^{3}+1}$
b. ${b}_{n}={\left(\frac{n+5}{n}\right)}^{n}$
c. ${c}_{n}={n}^{\frac{1}{n}}$

### Answer & Explanation

Bob Huerta

Beginner2022-01-07Added 41 answers

a. ${a}_{n}=\frac{3{n}^{3}}{{n}^{3}+1}$
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{3{n}^{3}}{{n}^{2}+1}$
$=\underset{n\to \mathrm{\infty }}{lim}\frac{3}{1+\frac{1}{{n}^{3}}}$
$=\frac{3}{1+0}$
$=3$
b.${b}_{n}={\left(\frac{n+5}{n}\right)}^{n}$
$\underset{n\to \mathrm{\infty }}{lim}{b}_{n}=\underset{n\to \mathrm{\infty }}{lim}{\left(\frac{n+5}{n}\right)}^{n}$
$=\underset{n\to \mathrm{\infty }}{lim}{\left(1-\frac{5}{n}\right)}^{n}$
${e}^{5}$
c. ${c}_{n}={n}^{\frac{1}{n}}$
$\underset{n\to \mathrm{\infty }}{lim}{n}^{\frac{1}{n}}=1$

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