Is there a more general definition of a limit? I've been bothered with the standard definition of the limit in analysis for a while: 
'A sequence $(x_n)$ has $x$ as a limit if $\forall \epsilon>0\ \exists N\in\mathbb{N}$ s.t. $\forall n\geq N:\ |x_n-x|<\epsilon$.' 
I feel it is too restrictive. I am wondering if there is some more general definition of a limit, perhaps in terms of a probability measure. Is someone aware of such a kind of limit existing?
This question is mostly inspired by the Bolzano-Weierstrass theorem: 
'Every bounded sequence in $\mathbb{R}$ has a convergent subsequence'. Thus, in every bounded divergent sequence, there is(/are) some hidden limit(s), yet the sequence as a whole is said to have no limit. Could one perhaps define a limit in such a way that these converging subsequences are taken into account?
For example, consider the sequence $(0,1,0,1,0,1,\dots)$. What is the limit of this sequence? I am aware that one could use the Cesaro mean and say that the limit is $\frac{1}{2}$. However, intuitively this doesn't feel right. I'd say the limit is half 0 and half 1. When you consider an element arbitrarily far along in this sequence you lose track of whether you consider an even or an odd element, and you have a probability of $1/2$ of finding a 0 and a probability of $1/2$ of finding 1.
A continuous probability distribution might be a limit as well. I would say the limit of $x\mod 1$ as $x$ tends towards infinity is a uniform probability measure on $[0,1]$.
Is anyone aware of the type of limit I am trying to describe here?

The set of limit points/cluster points/accumulation points of a sequence will usually/always be the set you are trying to describe. Definitions vary depending on the source, but "limits of subsequences" probably captures it unambiguously when you're looking at sequences in the reals (or $\mathbb R^n$).
