How do the roots change How do the roots of the  quadratic  equation    $ax^2+bx+c=0$ change when $b$ and $c$  retain constant values and $a$ tends to zero?      ($b\neq 0$)

The roots are given (on any field of characteristic $\neq2$) by:
$$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Taking this as a function of $\;a\;$, we can apply L'Hôpital's rule for one of the roots:
$$\lim_{a\to0}\frac{-b+\sqrt{b^2-4ac}}{2a}=\lim_{a\to0}-\frac{c}{\sqrt{b^2-4ac}}=-\frac c{|b|}$$
assuming, of course, $\;b\neq0\;$.
For the other one the numerator doesn't vanish but the denominator does, so the limit is $\;\pm\infty\;$, depending on the sign of $\;b\;$.
