A typical boundary condition for an initial boundary value problem is $$ \lim_{x\rightarrow\infty} T(x,t) = T_\infty.$$
For example, this might be the temperature at the end of a very long rod. Under what conditions is it equivalent to instead enforce the boundary condition $$ \lim_{x\rightarrow\infty} \frac{\partial T}{\partial x} = 0$$
or, perhaps worded differently, is it ever the case, given $T\rightarrow T_\infty$ as $x\rightarrow\infty$, that $\partial T/\partial x\ne0$ as $x\rightarrow\infty$? The alternative boundary condition makes sense physically, but does there exist a formal mathematical treatment of this?