\lim_{x\to0}\frac{\frac{\beta^2x^3}{2}-\frac{\alpha^2x^3}{2}+o(x^2)}{(x+o(x))^2}=
\lim_{x\to0}\frac{\frac{\beta^2x^3-\alpha^2x^3}{2}+o(x^2)}{x^2+o(x^2)}=0