finding the limit of this sequence:
lim
n->infinity (cosn)/(sqrt(n))
While graphing, it looks like the limit is approaching 0.
however, how could i demonstrate this mathematically?
could I use l'hopitals rule?
thankyou.
First you need to know that as $\displaystyle n \to \infty, \sqrt{n} \to \infty$
$\displaystyle 0 \le |\cos(n)/\sqrt{n}| \le 1/\sqrt{n}$
hence as $\displaystyle n$ becomes large $\displaystyle |\cos(n)/\sqrt{n}|$ is trapped between zero and a number which becomes arbitarily small, and hence tends to zero. But if the absolute value of a sequence tends to zero so does the sequence itself.