let {xn} as n tends to infinity be defined by xn=cos(n)/n. The limit is 0, but is the only way to justify this to use the definition of a limit?
$\displaystyle -1 \leq cos(n) \leq 1 $
$\displaystyle \frac{-1}{|n|} \leq \frac{cos(n)}{n} \leq \frac{1}{|n|} $
as n tends to infinity, you can take n > 0
$\displaystyle lim_{n\to\infty}\frac{-1}{n} = 0 $ and $\displaystyle lim_{n\to\infty}\frac{1}{n} = 0 $
By the squeeze thereom $\displaystyle \lim_{n\to\infty}\frac{cos(n)}{n} = 0 $
Well, that is a bit of a more tricky situation. You can bound it above and below, but this isn't exactly helpful. Do you know that $\displaystyle \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=e$. If so, just split your limit into $\displaystyle \left(1+\frac{1}{n}\right)^n\left(1+\frac{1}{n}\ri ght)$....and then.............what?