Do polynomials have a limit ratio of 1 over fixed intervals? If $p$ is an increasing polynomial function, and $c>0$ is a constant, do we always have:
$$ \underset{x \rightarrow \infty}{\lim}  \dfrac{p(xc+c)}{p(xc)} = 1?$$
I think if I am not missing anything, as the coefficient and degree of the leading terms will be equal in the numerator and denominator, by L'Hôspital's rule the limit should be equal to $1$. I am wondering if there is an easier proof for this fact, or anything that I am missing.

It suffices to show that for any non zero polynomial with degree $n$
$$\lim_{x\to \infty}\frac{p(x)}{x^n}=a_n$$
then
$$\lim_{x\to \infty}\frac{p(xc+c)}{p(xc)}=\lim_{x\to \infty}\frac{p(xc+c)}{(xc+c)^n}\frac{(xc)^n}{p(xc)}\frac{(xc+c)^n}{x^n}\frac{x^n}{(xc)^n}=\frac{a_nc^n}{a_nc^n}=1$$
