Originally Posted by

**lllll** I'm having a lot of trouble showing that that the $\displaystyle \lim_{x\rightarrow \infty} e^x > x^p \ \forall \ p \ \in \ \mathbb{R}$

I figure it has to be done by induction, so if I wanted to show that $\displaystyle e^x $tends to infinity faster then $\displaystyle x^p$ I would have:

$\displaystyle \lim_{x\rightarrow \infty} e^x > x^1$ which is obviously true

so assume that

$\displaystyle \lim_{x\rightarrow \infty} e^x > x^p$ is true

then I would have $\displaystyle \lim_{x\rightarrow \infty} e^x > x^{p+1}= e^x>x^p\cdot x =\frac{e^x}{x}>x^p$ where the inequality holds only if $\displaystyle e^x $ tends to infinity at a faster rate then $\displaystyle x^p$