For every $n$, I have a polynomial $p_n(x)=a^{(n)}_{n-1}x^{n-1}+a^{(n)}_{n-2}x^{n-2}+\dots+a^{(n)}_0$ (the $n$ in the exponent of the coefficients is merely an index).
I can show that $\lim_{n\to\infty}\sqrt[n]{a^n_{n-1}}=C$ for some constant C, and that this sequence is rising.
I can also show that we have $\lim_{n\to\infty}\sqrt[n]{p_n(x)}=A_x$ for some constant $A_x$.
My goal is to show that $\lim_{x\to\infty}(A_x/x)\ge C$.
I cannot assume the limits (of n and x) can be interchanged, although if it's easily provable I'll be glad to hear how.
The major obstacle I fail to see how to tackle is the fact that we take the $n$-th root, but the polynomial is of degree $n-1$. Were the polynomial of degree $n$ it would be much more natural as the division by $x$ would cancel out exactly with the leading coefficient of the polynomial.
Note that the claim may be incorrect (though it's unlikely) or I might be missing some assumptions (much more likely).