To find the limit
\[
\lim_{x \to 0} \frac{x}{\left( \frac{6}{x+4} \right) - \left( \frac{3}{x+2} \right)}
\]
we start by simplifying the denominator. The expression in the denominator is:
\[
\frac{6}{x+4} - \frac{3}{x+2}
\]
To combine these fractions, we need a common denominator, which is \((x+4)(x+2)\):
\[
\frac{6(x+2) - 3(x+4)}{(x+4)(x+2)}
\]
Now, simplifying the numerator:
\[
6(x+2) = 6x + 12
\]
\[
3(x+4) = 3x + 12
\]
Thus, the


