$\mathop {\lim }\limits_{x \to 3} \frac{x^3-2\left| {x+3} \right|+3\left| {x-4} \right|-18}{2x^2-3x-9}=\mathop {\lim }\limits_{x \to 3}\frac{x^3-2(x+3)-3(x-4)-18}{(x-3)(2x+3)}$$=\mathop {\lim }\limits_{x \to 3}\frac{(x-3)(x^2+3x+4)}{(x-3)(2x+3)}=\mathop {\lim }\limits_{x \to 3}\frac{x^2+3x+4}{2x+3}$
$=\frac{22}{9}$