Finding relation between coefficients in a limit I was practicing questions and got stuck at this. There is no solution provided in the book. What I guessed is that we will be using the properties of discriminant of a quadratic equation and draw the graphs and choose an $x_0$, but I do not know how to actually proceed with it.
Any help would be appreciated.


Let $\alpha, \beta$ be the roots of $a x^2+b x+c=0,$ where
  $1<\alpha<\beta .$ 
Then $\lim\limits_{x\to x_0} \frac{\left|ax^2+bx+c\right|}{a x^2+b x+c}=1$ then which of
  the following statements is incorrect
(a) $a>0$ and $x_0<1$
(b) $a>0$ and $x_0<\beta$
(c) $a<0$ and $\alpha<x_0<\beta$
(d) $a<0$ and $x_0<1$


https://undergroundmathematics.org/quadratics/r5138/images/q5138.png
See this graph for a>0 (The labeling of the graph is incorrect, and I can't find another one)
Those points are the roots. first one is $\alpha$ and second one is $\beta$. Both are greater than 1
$$\lim\limits_{x\to x_0} \frac{\left|ax^2+bx+c\right|}{a x^2+b x+c}=1$$
Forget the modulus, without the modulus, the limit is always 1, provided $x_0≠ \alpha,\beta$. But since modulus is there, the limit can be -1 too.
HINT:
If $a>0$ and $x_0>\beta$, both numerator and denominator will be positive.
now you can check all the options, I suppose.
