How to compute the limit of this difference of the n'th root of polynimials?

Question

From playing around, it appears to me that

$$a_n=b_n\implies\mathrm{lim}_{x\to\infty}\left(\left(\sum_{k=0}^n a_k x^k\right)^\frac{1}{n}-\left(\sum_{k=0}^n b_k x^k\right)^\frac{1}{n}\right)=\frac{a_{n-1}-b_{n-1}}{n\ a_n^\frac{n-1}{n}}.$$

For example

$$\mathrm{lim}_{x\to\infty}\left(\sqrt{2+81x+3x^2}-\sqrt{5+40x+3x^2}\right)=\frac{41}{2\sqrt{3}}.$$

How to proof this?

(I have an approach, with lots of half backed steps: Taking the expression and plugging in $x=\frac{1}{y}$, I can make a "series expanstion at $x=\infty$ by considering the expanstion at $y=0$" and this way the right coefficients already pop up.)

Answer 1

If $p(x) = \sum_{k=0}^n a_k x^k$ with $a_n > 0$, then $p(x) = a_n x^n \left(1 + \dfrac{a_{n-1}}{a_n x} + O(1/x^2)\right)$ as $x \to \infty$, and so $p(x)^{1/n} = a_n^{1/n} x \left(1 + \dfrac{a_{n-1}}{n a_n x} + O(1/x^2)\right)$.