Let us consider the functors from dg algebras to graded spaces given by Hochshcild homology and cyclic homology. Then it is well known that in degree $0$, both functors are given by $A\longmapsto A/[A,A]$. However, if $A$ is a usual associative algebra over a field of characeristic $0$, and if we pick a free model $B=(TV,d)\longrightarrow A$, the homology of $B/[B,B]$ is $\mathrm{HC}_*(A)$. In this sense, HC is the right Taylor expansion for the abelianization functor in characteristic zero. At some point I naïvely thought that this would give HH instead. This motivates the following questions:

Q1. How should I have known that taking the left derived functor of abelianization in the model category $\mathsf{Alg}$ would give HC and not HH, even though at $t=0$ both theories coincide?

Q2. What other examples are there of this phenomenon? (If the example s in $\mathsf{Alg}$, better!)

Q3. Can one describe the functors that coincide with HC and HH at $t=0$ in the sense above, but nonetheless give different theories in large degrees? Or at least give examples, as in Q2?