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.