I'm trying to understand a definition of a positive definite function I found in a book on Lyapunov stability
Definition: A continuous function $W(x)$ is said to be a positive definite function if $W(0) = 0$, and $W(x) > 0$ in $R^n$ and there exists some $r > 0$ such that
$\underset{||{x}|| > r }{\text{inf}} (W(x) > 0)$
What does the last condition exactly mean? I don't fully understand the infimum but I usually think of it as a minimum value.
Later an example is given of a positive definite function, e.g. $x^2$. But $x^2/(1+x^4)$ is said not to be positive definite? How can I deduce this from the definition? Does it have to do with the fact that the function doesn't go to infinity as $x$ goes to infinity?