If  $f\left(x\right)$ is a continuous function , then $\underset{x\to 0}{\lim }f\left(x\right)\frac{\left|x\right|}{x}$ exist  if

$\underset{x\to 0^{+}}{\lim }\frac{\left|x\right|}{x}=1$ and $\underset{x\to 0^{-}}{\lim }\frac{\left|x\right|}{x}=-1$ also $f\left(x\right)$ is continuous

$\therefore$ Given limit can exist only if $\underset{x\to 0}{\lim }f\left(x\right)=0$

$\therefore f\left(x\right)=a{x}^2+bx$ is the only choice