A few things:
1) The usual notation for the field of rational functions over a field is to use parentheses, so the field you're looking for is denoted $\mathbb{F}_p(x)$.

2) The field of rational functions isn't the extension of $\mathbb{F}_p[x]$ you probably want, as it includes $x^{-1}$, which can't be limited to in your sense; the extension you want probably should be $\mathbb{F}_p[[x]]$, the ring of formal power series $\sum_i a_i x^i$ over $\mathbb{F}_p$. This ring has a valuation, allowing the concept of limits to make sense. The term you may want would be "valuation ring". And in this ring, $(1-x) \sum x^i = 1$. 

3) There is a field $\mathbb{F}_p((x))$ extending both; it is the field of fractions of $\mathbb{F}_p[[x]]$. One way to think of it is as the formal power series "starting" at some minimum power of $x$. Every rational function can be expressed as an element of this field.