Question

If $l, m$ $(l<m)$ are roots of $ax^2 + bx + c = 0$, then $\lim_{x \to a} \left| \dfrac{ax^2 + bx + c}{ax^2 + bx + c} \right|$ =

• A. $\dfrac{|d|}{a},\ \forall\ \alpha \in \mathbb{R}$
• B. $\dfrac{-|d|}{a},\ \alpha \notin (l,m)$
• C. $\dfrac{-|d|}{a},\ \alpha \in (l,m)$
• D. $\dfrac{|d|}{a},\ \alpha \in (l,m)$

Hint:

Sign of quadratic between roots is opposite of leading coefficient; watch open intervals carefully.

Answer 1

The Correct Option is C

Let $f(x) = ax^2 + bx + c$ with roots $l, m$
Then sign of $f(x)$ between $l$ and $m$ depends on sign of $a$
Since $f(x) = a(x - l)(x - m)$, in $(l, m)$, $(x - l)(x - m)<0$ ⇒ sign opposite to $a$
So limit as $x \to \alpha$ inside $(l,m)$ gives negative sign ⇒ $-|d|/a$