Step 1: Factor the quadratic.
Since $l$ and $m$ are the roots of $ax^2+bx+c=0$, we can write $ax^2+bx+c = a(x-l)(x-m)$ with $l < m$.
Step 2: Recall what the ratio measures.
The expression $\dfrac{|ax^2+bx+c|}{ax^2+bx+c}$ equals $+1$ where the quadratic is positive and $-1$ where it is negative. So it is the sign of the quadratic.
Step 3: Determine the sign inside $(l,m)$.
If $\alpha \in (l,m)$, then $\alpha - l > 0$ and $\alpha - m < 0$, so $(\alpha-l)(\alpha-m) < 0$. Hence $a(\alpha-l)(\alpha-m)$ has the opposite sign to $a$.
Step 4: Translate sign into the ratio.
When the quadratic has sign opposite to $a$, the ratio $\dfrac{|\cdot|}{\cdot}$ equals $-\dfrac{|a|}{a}$, because $\dfrac{|a|}{a}$ records the sign of $a$.
Step 5: Note continuity gives the limit.
Throughout the open interval $(l,m)$ the quadratic keeps a constant sign, so the limit as $x \to \alpha$ for any $\alpha$ inside is just that constant value $-\dfrac{|a|}{a}$.
Step 6: State the matched option.
Therefore the limit is $-\dfrac{|a|}{a}$ when $\alpha \in (l,m)$.
\[ \boxed{-\dfrac{|a|}{a},\ \alpha\in(l,m)} \]