Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $ω_ε = (x_0 − ε, x_0 + ε)$, in the limit $ε → 0$, where $x_0 ∈ (0, 1)$. It is known that depending on arithmetic properties of $x_0$ , there may exist a minimal time $T_0$ of pointwise control at x_0 of the heat equation. Besides, for any ε fixed, the heat equation is controllable with control set $ω_ε$ in any time $T > 0$. We relate these two phenomena. We show that the observability constant on $ω_ε$ does not converge to $0$ as $ε → 0$ at the same speed when $T > T_0$ (in which case it is comparable to $ε 1/2$) or $T < T_0$ (in which case it converges faster to $0$). We also describe the behavior of optimal $L^2$ null-controls on $ω_ε$ in the limit $ε → 0$.