## 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 \[x \rightarrow \infty\]

.