Question

$\lim_{x\to 0} \frac{(a+b)+cx-bx^{2}/2}{x^{2}}$

Answer 1

The limit is $-\frac{b}{2}$, provided that $a+b=0$ and $c=0$.

Answer 2

The limit is of the form $\frac{a+b}{0}$ as $x \to 0$. For a finite limit, $a+b$ must be 0. The expression then simplifies to $\lim_{x\to 0} (\frac{c}{x} - \frac{b}{2})$. For this to be finite, $c$ must be 0. With $a+b=0$ and $c=0$, the limit is $-\frac{b}{2}$.