After struggling with this limit for a while, I would really, really appreciate your help with solving this problem.

Let $\displaystyle a, b, c \in \mathbb{R}_+$ be positive real numbers.

*Without* using the l'Hôpital's rule, find the limit

$\displaystyle \lim_{x\to 0}

\left( \frac{a^{x^3}+b^{x^3}+c^{x^3}}

{a^{x^2}+b^{x^2}+c^{x^2}} \right) ^\frac{1}{x^2}$

I tried every method of attack known to me, but to no avail; substitution; trying to utilize the identity $\displaystyle \lim_{x\to 0} (1+x)^\frac{1}{x}=e$, ... nothing seemed to work.

Mathematica says the limit should be $\displaystyle \frac{1}{\sqrt[3]{abc}}$, but I have no idea how to arrive at that.

I'd be very grateful for any help.