certain cases, so define this special one but frankly it doesn't make my brain happy (but who am I)
It's not a special case. It's the standard general definition of the limit of a sequence. The limit of a sequence a_1,a_2,...,a_n,... exists if and only if there exists some L (the limit) such that, for any (arbitrarily small) epsilon > 0, there exists some N with the property |a_n - L| < epsilon for every n>N. In this case a_n is n 9s after the decimal point (or a_n = 9*sum_{i=1}^n (.1)^i).
Technically it’s an infinite series (sum of an infinite sequence), which is a finite number in certain cases like this one.