Note that:
$$ \lim_{S\rightarrow \infty} C(t,S) =S-K{\rm e}^{-r(T-t)} $$
for all $t$.
Basically because one can easily accept
$$ \lim_{S\rightarrow \infty} P(t,S) =0 $$
and one still expects the put-call parity to hold:
$$ \lim_{S\rightarrow \infty} C(t,S) - \lim_{S\rightarrow \infty} P(t,S) =S-K{\rm e}^{-r(T-t)} $$