\[e^x\]
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 \[x \rightarrow \infty\]
.