If we're talking about any possible system and what it potentially yields to the most competent person, then logic is virtually unlimited in its application. The only area where I can think of it not applying is when value judgments come up. We use logic to apply values to reality, but generally, all values have an illogical foundation. There is no logic involved in your feeling that a cheeseburger is or isn't tasty.
However, all human thoughts and feelings are just a biological process, which means all of your values are probably derived from a set of switches, of sorts. In such a case, it means there is a system to those switches, and the values they produce, that could be understood through logic. So maybe even values are within logic.
However, our experience of values, as well as many of our other feelings and senses, can probably never be understood logically. Alas, logic itself is an abstract construct of ours, and something we sort of experience. This brings about an issue with the OP's question. How are we supposed to understand the word limited? I say logic is applicable to anything, but it does not explain everything about what it is applied to.
As for whether or not there is truth outside of logic, I'd say no. True and false are the most basic components of logic. Even acknowledging your illogically formed values requires you to make a statement about them within the real of true/false.