granica ciagu

[saguarowega]

1.\(n^ \frac{2}{3}[(n+1)^ \frac{1}{3} -n^ \frac{1}{3}]\)
2.\(\frac{nsin(1/n)}{n \sqrt{n}+2 }\)

[octahedron]

\(n^{\frac{2}{3}}\left[(n+1)^{\frac{1}{3}}-n^{\frac{1}{3}}\right]=n^{\frac{2}{3}}\left[\frac{(n+1)-n}{(n+1)^{\frac{2}{3}}+n^{\frac{1}{3}}(n+1)^{\frac{1}{3}}+n^{\frac{2}{3}}}\right]=\frac{1}{\left(\frac{n+1}{n}\right)^{\frac{2}{3}}+\left(\frac{n+1}{n}\right)^{\frac{1}{3}}+1}\stackrel{n\to\infty}{\to}\frac{1}{1^{\frac{2}{3}}+1^{\frac{1}{3}}+1}=\frac{1}{3}\)

[octahedron]

\(-\frac{n}{n\sqrt{n}+2}<\frac{n\sin\left(\frac{1}{n}\right)}{n\sqrt{n}+2}<\frac{n}{n\sqrt{n}+2}\\
\frac{n}{n\sqrt{n}+2}=\frac{1}{\sqrt{n}+\frac{2}{n}}\stackrel{n\to\infty}{\to}\frac{1}{\infty+0}=0\quad\Rightarrow\quad\frac{n\sin\left(\frac{1}{n}\right)}{n\sqrt{n}+2}\stackrel{n\to\infty}{\to}=0\)