1. ## A fiendish limit

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

Let $a, b, c \in \mathbb{R}_+$ be positive real numbers. Without using the l'Hôpital's rule, find the limit

$\lim_{x\to 0}
\left( \frac{a^{x^3}+b^{x^3}+c^{x^3}}
{a^{x^2}+b^{x^2}+c^{x^2}} \right) ^\frac{1}{x^2}$

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

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

I'd be very grateful for any help.

2. Originally Posted by gusztav
After struggling with this limit for a while, I would really, really appreciate your help with solving this problem.

Let $a, b, c \in \mathbb{R}_+$ be positive real numbers. Without using the l'Hôpital's rule, find the limit

$\lim_{x\to 0}
\left( \frac{a^{x^3}+b^{x^3}+c^{x^3}}
{a^{x^2}+b^{x^2}+c^{x^2}} \right) ^\frac{1}{x^2}$

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

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

I'd be very grateful for any help.
In the first instance, the special case a = b = c is simple enough and the answer agrees with Mathematica.