# Proving that exponential is growing faster than polynomial

I want to prove this limit:$$lim_{x to +infty}(|P(x)|-e^x)=-infty$$.
I can’t use Hopital’s rule, but only the fact that $$a_{n}=(1+frac{x}{n})^n$$ and $$b_{n}=(1+frac{x}{n})^{n+1}$$ converge to $$e^x$$.