sanuluy

2021-08-22

Find the limits of the following sequences

Theodore Schwartz

Skilled2021-08-23Added 99 answers

We have, by applying the Squeeze Theorem

$\underset{n\to \mathrm{\infty}}{lim}\frac{\mathrm{sin}\left(n\right)}{n}=0$

We have

$\underset{n\to \mathrm{\infty}}{lim}\frac{n\cdot \mathrm{ln}\left(n\right)}{{n}^{2}+1}$

$=\underset{x\to \mathrm{\infty}}{lim}\frac{\frac{\mathrm{ln}\left(n\right)}{n}}{1+\frac{1}{{n}^{2}}}$

This implies

$=\frac{\underset{n\to \mathrm{\infty}}{lim}\frac{\mathrm{ln}\left(n\right)}{n}}{\underset{n\to \mathrm{\infty}}{lim}1+\frac{1}{{n}^{2}}}$

Since$\underset{n\to \mathrm{\infty}}{lim}\frac{\mathrm{ln}\left(n\right)}{n}=0$
$=\frac{0}{1}=0$

We have

This implies

Since

22+64

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