To begin with, I think it should be noted that
- $b_i=0$ "stops" the continued fraction in its track, so it would probably be best to ignore $0$, except perhaps for $b_0$.
- also, the final term $b_n$ better not be $-1$ (I don't know about you, but I really don't like going anywhere near the divide-by-zero monster)
With that out of the way, I can't say that I clearly understand the notion of limit in this context. A finite continued fraction simply results in a number. If we're interested in upper/lower bounds on what numbers can be represented with this set of rules, a couple examples seem to suggest that there aren't any. For instance with
$$
[-1,b_1] = 1-\cfrac{1}{1+b_1} = \cfrac{b_1}{1+b_1}
$$
it is clear that the closer to $-1^+$ we choose $b_1$, the more negative the value will become, tending to $-\infty$. Similarly, with
$$
[-1,-1,b_2] = 1-\cfrac{1}{1-\cfrac{1}{1+b_2}} = -\cfrac{1}{b_2}
$$
it is also clear that the closer $b_2$ approaches $0^-$, the larger the value gets, tending to $+\infty$ this time.