= Solution
As $\sigma\downarrow0$, the <Black-Scholes formula> tends to $\boxed{(S_0-Ke^{-rT})^+}$. This also follows by discounting the deterministic zero-volatility payoff; at equality $S_0=Ke^{-rT}$ the limiting value is zero.
If $S_0>Ke^{-rT}$ and a call trades at $C<S_0-Ke^{-rT}$, buy a call, short one share, and buy a bond paying $K$ at maturity. Invest the positive initial surplus $S_0-Ke^{-rT}-C$. At maturity the call, share and bond together pay $(S_T-K)^+-S_T+K=(K-S_T)^+\geq0$, in addition to the strictly positive invested surplus. If the lower bound is zero, violating it means $C<0$: buy the call and invest the cash received, obtaining a nonnegative option payoff and a strictly positive bank payoff. Both strategies are <arbitrages>.
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