Consider a polynomial in $r$ of the form $$ r^4+p_3(\lambda)r^3+p_2(\lambda)r^2+p_1(\lambda)r+p_0(\lambda), $$ where the $p_i$ are polynomials in the parameter $\lambda$. I use degree four to simplify the notation but you can think of a polynomial of any degree you want. Then I believe this result to be true: as $\lambda\to\infty$, the zeros of the polynomials above approach the solutions to $$ r^4+l_3 r^3+l_2r^2+l_1r+l_0=0, $$ where $l_i$ is the leading term of $p_i$.
For example, I would like to be able to say that the solutions of $$ r^4+5r^3+(\lambda+1)r^2+(6\lambda+5)r+17\lambda^2+3=0,\;\;\;\;(1) $$ approach the solutions to $$ r^4+5r^3+\lambda r^2+6\lambda r+17\lambda^2=0 $$ as $\lambda\to\infty$. One thing I can do is to apply the scaling $r=\sqrt{\lambda}\rho$ and substitute in (1), divide by $\lambda^2$ and apply the limit $\lambda\to\infty$ to obtain $$ \rho^4+\rho^2+17=0,\;\;\;\;\;(2) $$ which seems to show that the solutions to (1) approach the quantities $\sqrt{\lambda}\rho$, where $\rho$ are the solutions of (2). It does not contradict the result I am trying to prove and one would think it is possible to apply this scaling argument all the time. However, the advantage of the result I want to prove is that it is very easy to state and apply and does not require finding the correct scaling for each particular case.
Of course, if one would take the limit as $\lambda$ approaches a finite real value, then the solutions approach the solutions of the polynomial where the limit is applied to each coefficient. However, the difficulty here is that the limit is at $\infty$ and I could not find any results concerning this type of problems. Any reference would be appreciated.