Recall that profit maximization occurs at MR=P(1-1/|E|)=MC, whether in monopolistic or competitive pricing.
Thus a more accurate statement would be:"a monopolistic firm charge when |E|>1 and competitive firm charge at |E|→∞."
Think: When |E|<1, the monopolist can always raise his price to increase revenues, which implies that |E|<1 is not a profit maximizing portion. Also, any competitor faces a flat demand, ie. |E|→∞.