Could somebody please show how $$\lim_{n \to \infty} x^n = 0$$ I always think that considering $|x|>1$ as $n$ becomes larger, $x^n$ also becomes larger. In this case,$$\lim_{n \to \infty} x^n = \infty$$ Could somebody please explain where I've gone wrong?
Thanks very much!
This is clearly not true always.
It is only true when $|x|\lt 1$, otherwise, it will go to infinity as you said.
Unless x=1, then it will just stay as 1 , as pointed out below.
Ie,
$\lim_{n \to \infty}x^n= 0$ if $|x| \lt 1$
$\lim_{n \to \infty}x^n= 1$ if $x=1$
$\lim_{n \to \infty}x^n= \infty$ if $|x| \gt 1$