oliviayychengwh

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.$c}_{n}={n}^{\frac{1}{n}$

a.

b.

c.

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.$b}_{n}={\left(\frac{n+5}{n}\right)}^{n$

$\underset{n\to \mathrm{\infty}}{lim}{b}_{n}=\underset{n\to \mathrm{\infty}}{lim}{\left(\frac{n+5}{n}\right)}^{n}$

$=\underset{n\to \mathrm{\infty}}{lim}{(1-\frac{5}{n})}^{n}$

$e}^{5$

c.$c}_{n}={n}^{\frac{1}{n}$

$\underset{n\to \mathrm{\infty}}{lim}{n}^{\frac{1}{n}}=1$

b.

c.

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