Let
$$E:=\overbrace{-2C_0-(n-2)C_1}^A+(4+(n-2)\overbrace{(2C_1+C_2)}^B)\cdot p-(n-2)\overbrace{(C_1+C_2-C_3)}^Cp^2\\$$
I got a question when I tried to give asymptotics analysis of $E^2$. It looks contradiction to the computation obtained in mathematica, which pretty confused me...
I am analyzing the asymptotics of the function $E^2$. where $p=p(n)$ is a function of $n$, and it is asymptotically equivalent to
$$p(n)\sim \frac{2C_1+C_2}{2(C_1+C_2-C_3)}-\frac{\sqrt{C_2^2+4C_1C_3}}{2(C_1+C_2-C_3)} + K\cdot g(n)$$
where $0 < \epsilon \ll 1$, $g(n)$ is a decreasing function, $$K:=\frac{2c_2^2+8c_1c_3+(-4c_1+2c_0c_1-2c_2+2c_0c_2-2c_0c_3) \sqrt{c_2^2+4c_1c_3}}{(c_1+c_2-c_3)(c_2^2+4c_1c_3)}>0$$
From my deduction, $E\sim n^2$ (the deduction is long so I put it here), however, from the mathematica, it gives $E\sim n^2 g^2(n)$.
What could go wrong? Thank you very much in advance for any help!