For all (fixed) natural numbers $n$, we have that $$\lim_{x\to\infty}\frac {\exp(x)}{\exp_x(n)}=\lim_{x\to\infty}\frac {e^x}{x^n}=\infty,$$ $$\lim_{x\to\infty}\frac {\exp_x(n)}{\exp(x)}=\lim_{x\to\infty}\frac {x^n}{e^x}=0.$$ These limits show that if $x$ is growing, the growth of the exponential function of $x$ is faster than the growth of any $n$-th power of $x$, no matter how big $n$ is.
Proofs: 1
