Question

Show that when $n$ is infinite the limit of $n{x}^n$ tends to $0$, when $x>1$.

Answer 1

Let $x=\frac{1}{y}$ so that $y>1$

Also, let $y^n=z$

$\Rightarrow n\log y=\log z$

Then,

$n{x}^n=\frac{n}{y^n}=\frac{1}{z}.\frac{\log z}{\log y}=\frac{1}{\log y}.\frac{\log z}{z}$

Now, when $n$ is infinite, $z$ is infinite and

$\frac{\log z}{z}=0$

Also, $\log y$ is finite

Therefore, $\underset{n\to \infty }{lim}n{x}^n=0$