Derived Limits vs Limits in the Derived Category

This question is more or less a reference request. Assume $A$ is a (Grothendieck) abelian category, and we consider its derived category, and we know the inverse limit in $A$ is left exact, and we have its right derived $R\varprojlim$. On the other hand, we also have $\varprojlim$ in $D(A)$ (although as Zhen has remarked below, $D(A)$ is not complete). My question is basically the following: in what extent, are these two things coincide?

Of course, these two definitions are not applied to exactly the same situations. If we have $J$ a small category of indices, and we have an abelian category $Fun(J,A)$, and $R\varprojlim_J$ is a functor from $D(Fun(J,A))$ to $D(A)$. (Of course there are other limit systems in $D(A)$ that is not of the form like this). We have for example both $$ R\varprojlim_J R\underline{Hom}(M^\bullet, N^\bullet_j)=R\underline{Hom}(M^\bullet, R\varprojlim_J N^\bullet_j)$$ Is there some special cases when $R\varprojlim N_j^\bullet$ is exactly the inverse limit that we want.

The situation that interests me most is when $J\to D(A)$ actually comes from a limit system $N:J\to Ch(A)$, and I want to know whether $R\varprojlim_j N_j^\bullet=\varprojlim_j N_j^\bullet$. (By the way, I am not sure whether $Ch(A)\to D(A)$ preserves limit in some good situation)

I actually doubt there might be some subtlety in this problem. If the two are not the same in general, which one is the notion that is used more often, and is there some philosophical reason for that?

Any comment is sincerely welcomed!

There are essentially no limits in the derived category, whether the diagram factors through chain complexes or not. For instance, any monomorphism in any triangulated category must be split. This is a nice exercise from the axioms. All that exists is products and coproducts, almost without exception. One perspective on the purpose of derived limits is precisely to repair this defect of the derived category as well as possible.