The librarian said: The students may not borrow more than 3 books in one month.
So, Is there a possibility for students to borrow 4 books per month? and why?
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The usage of 'may' as an order is archaic and not commonly used except by royalty. 'May this be done.' is a complete sentence and an example of Imperative mood, but not used often in everyday language.
For more information of other possible usages, there is a good discussion in wikipedia of usages of May and Might.