To Infinity But Slightly Less...

Maybe this is what happens inside a thermometer where the temperature charged material flows with the temperature field until $V=0$,

from the previous post "Only If You Started Somewhere" dated 27 Dec 2017.  In this case, $r\ne\infty$ at $V=k\Delta T=0$.

Note:  Mercury has a thermal expansion of $60.4\,\mu m\, (m^{-1}k^{-1})$ at $25\,^oC$, that means a thermometer of one meter will expand by $60.4 \,\mu m$ per Kelvin increase in temperature.