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:

$$ \lim_{S\rightarrow \infty} C(t,S) - \lim_{S\rightarrow \infty} P(t,S) =S-K{\rm e}^{-r(T-t)} $$