little-o and its properties

I know that $f(x) = o(g(x))$ for $x \to \infty $ if (and only if) $\lim_{x \to \infty}\frac{f(x)}{g(x)}=0$ Which means than $f(x)$ has a order of growth less than that of $g(x)$.

1) I'm still confused if $x \to 0$. Because in this case $x^5 = o(x^2)$

2) Can someone list me the properties of little-o? For now, I know the following:

$f(x)*o(g(x) = o(f(x)*g(x))$

$o(f(x)) \pm o(f(x)) = o(f(x))$

Thank you!

Thanks to Wikipedia** here are some properties:

• $ o(f) + o(f) \subseteq o(f) $
• $ o(f) o(g)\subseteq o(fg) $
• $ o(o(f)) \subseteq o(f) $
• $ o(f) \subseteq O(f) $

** http://en.wikipedia.org/wiki/Big_O_notation#Little-o_notation

Also, the following document may help you:

http://www.math.caltech.edu/~2010-11/1term/ma001a1/bigolittleo.pdf