Example 2

Let $V$ be a normed space over a non-discrete valued field $K$, and let $f(x) = a_0 x^n + a_1 x^{n-1} + \cdots + a_n$ be a polynomial in the variable $x \in K$, with coefficients in $V$, such that $a_0 \neq 0$. For every vector $b \neq 0$ one has $f(x) \simeq b x^n$ as $|x|$ tends to $+\infty$.