$ \lim_{x \rightarrow \infty} e^{x} - x^{k} = + \infty , \forall k \geq 1 $?

Question

I was wondering if this was true. My logic was:

1. At x=0, $e^{x}$ is always greater than $x^{k}$, for $k \geq 1$.
2. The kth derivative of $e^{x}$ is $e^{x}$.
3. The kth derivative of $x^{k}$ is 1.
4. For $x \geq 1$, the kth derivative of $e^{x}$ will be greater than the kth derivative of $x^{k}$.
5. Therefore, the (k-1)th derivative of $e^{x}$ will eventually be greater than the (k-1)th derivative of $x^{k}$.
6. Thus, the same logic applies to the (k-2)th derivative of $e^{x}$ eventually being greater than the (k-2)th derivative of $x^{k}$.
7. This pattern goes on until the 1st derivative of $e^{x}$ is greater than the 1st derivative of $x^{k}$, meaning that $e^{x}$ will eventually surpass $x^{k}$, proving that the limit holds.

Is my reasoning correct?

Thanks in advance!