just tell me please if my idea correct is
$$\exp(x) = \sum_{k=0}^{n} \frac{x^k}{k!} > \frac{x^n}{n!} $$
let $x > R= (n+1)!$ $\Rightarrow $ $$\frac{x^n}{n!}>\frac{R^n}{n!} >\frac{R}{n!}>\frac{(n+1)!}{n!}= n+1 $$
let $n_{0} \in \mathbb{N}$ so that n>$n_{0}$ for a big $n_{0}$
for all M >0 let s choose R = M!
then for all $x > R =(n+1)!$ there is a $M =(n+1)$ so that $\exp(x) > M$
$ \Rightarrow \lim_{x \to \infty}\exp(x) = \infty $
is that correct ?