Why e^x Tends to Infinity Faster Than Any Power of x

We can write
$e^x$
as the sum of powers of
$x$
.
$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+...$

This means that
$e^x$
increases faster than any power of x, since this expression for
$e^x$
adds multiples of all powers of
$x$
.
This means that as
$x \rightarrow \infty, , \: \frac{e^x}{x^n} \rightarrow \infty$
for any value of
$n$
.
In fact if
$p(x)$
is any polynomial
$\frac{e^x}{p(x)} \rightarrow \infty$
as
$x \rightarrow \infty$
.