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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  
\[x \rightarrow \infty\]
.

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