# Sequence $x_n=(1-\frac{1}{n})\sin (n\pi/3)$lim sup, lim inf

Let $x_n=(1-\frac{1}{n})\sin (\frac{n\pi}{3})$ for n $\ge$1. denote l=limit inferior and s=limite superior Then

1. -$\sqrt{3}/2\le l\lt s \le \sqrt{3}/2$

2. -$1/2\le l\lt s \le 1/2$

3. $l=-1$,$s=1$

4. $l=0=s=0$

My attempt: I manually plotted the sequence terms for n=1,2,... for few n. And then saw that this sequence has 3 limit points namely -$\sqrt{3}/2$, 0 , $\sqrt{3}/2$. So least one is limit inf , largest of limit point is limit sup. But I need suggestions for more efficient way. Thanks

Hinit: First, simplify the expression $\sin\left(\frac{n\pi}{3}\right)$. Then, it will be much easier to
• @BigBang Calculate the term for $n=1,2,3,4,5$. Can you see a pattern? – 5xum Feb 5 '15 at 9:53