Let $F$ be a category cofibered in groupoids over category $C$. Given a morphism $x'\to x$ in $F$ lying over a morphism $A′\to A$ in $C$, there is an induced homomorphism $\operatorname{Aut} A'(x')\to \operatorname{Aut} A(x)$.
The kernel $\operatorname{Inf}(x'/x):=\operatorname{ker}(\operatorname{Aut} A'(x') \to \operatorname{Aut} A(x))$
is also called the group of infinitesimal automorphisms. That's the defition from stacks project
Question: What is the origin and motivation of the name 'infinitesimal' here? How is it connected to the naive geometrical/analytic intuition of infinitesimality (of course if there exist a connection at all)?