Fractional part always diverge right?

Let $f_n(x)$ be an $n$ -th degree polynomial with all real coefficients and a positive leading coefficient.

For a fixed $n$ , if $\displaystyle \lim_{x\to\infty} \left\{ \sqrt[n]{f_n(x)} \right\}$ converge, what can we say about the leading coefficient of $f_n$ ?

Details and Assumptions :

• $\{x\}$ denote the fractional part of $x$ .

Bonus : Denote the polynomial as $f_n(x) = a_n x^n +a_{n-1} x^{n-1} + \ldots + a_0$ . Prove that we can find the limit in terms of $a_n$ and $a_{n-1}$ .