Is there a useful notion of being “approximately computable”
If the family function $f(x,n)=f_n(x)$ is computable then these are exactly the $\Delta^0_2$ functions, or equivalently, the functions that are Turing reducible to the halting set $0'$, which are very well studied within computability-theory.

There are also other, not equivalent, notions of almost computable that go by names such as generically computable and coarsely computable, see e.g.
Jockusch et al.: Asymptotic Density and the Theory of Computability: A partial survey.