Given,

$l, m\left(l<m\right)$ are roots of $a{x}^2+bx+c=0$,

Also we know that value of function between the roots is negative or positive depending upon the value of $a$,

So, when $x\in \left(l,m\right)$ then $f\left(x\right)=a{x}^2+bx+c<0, \text{if} a>0$ and $f\left(x\right)=a{x}^2+bx+c>0, \text{if} a<0$

So now using the above concept we will solve the given limit,

$\underset{x\to \alpha }{\lim }\frac{\left|a{x}^2+bx+c\right|}{a{x}^2+bx+c}=$$\frac{-\left|a\right|}{a}$, when $\alpha \in \left(l,m\right)$