Marvin Mccormick

2021-08-21

Find the limits of the sequences
1)${\left\{\frac{{n}^{2}+3}{{n}^{3}+{n}^{2}-1}\right\}}_{n=1}^{\mathrm{\infty }}$
2)${\left\{n\mathrm{sin}\frac{\pi }{n}\right\}}_{n=1}^{\mathrm{\infty }}$

lamanocornudaW

1)
$\underset{n\to \mathrm{\infty }}{lim}\frac{{n}^{2}+3}{{n}^{3}+{n}^{2}-1}=\underset{n\to \mathrm{\infty }}{lim}\frac{{n}^{2}\left(1+\frac{3}{{n}^{2}}\right)}{{n}^{3}\left(1+\frac{1}{n}-\frac{1}{{n}^{3}}\right)}$
$\underset{n\to \mathrm{\infty }}{lim}\frac{{n}^{2}+3}{{n}^{3}+{n}^{2}-1}=\underset{n\to \mathrm{\infty }}{lim}\frac{\left(1+\frac{3}{{n}^{2}}\right)}{n\left(1+\frac{1}{n}-\frac{1}{{n}^{3}}\right)}=0$
$lim\frac{{n}^{2}+3}{{n}^{3}+{n}^{2}-1}=0$
2)
$\underset{n\to \mathrm{\infty }}{lim}n\mathrm{sin}\frac{\pi }{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{\mathrm{sin}\frac{\pi }{n}}{\frac{1}{n}}=\underset{n\to \mathrm{\infty }}{lim}\frac{\pi \left(\mathrm{sin}\frac{\pi }{n}\right)}{\left(\frac{\pi }{n}\right)}$
$=\underset{n\to \mathrm{\infty }}{lim}\frac{\pi \mathrm{sin}\frac{\pi }{n}}{\left(\frac{\pi }{n}\right)}$

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

Jeffrey Jordon