NOTE: I am self studying UML so I have nobody to verify my diagrams and hence I am posting here, so please bear with me. This is the problem I got from some PDF available on Google that simply had the following problem statement:

Problem Statement:
A library contains books and journals. The task is to develop a computer system for borrowing books. In order to borrow a book the borrower must be a member of the library. There is a limit on the number of books that can be borrowed by each member of the library. The library may have several copies of a given book. It is possible to reserve a book. Some books are for short term loans only. Other books may be borrowed for 3 weeks. Users can extend the loans.
1. Draw a use case diagram for a library.
2. Give a use case description for two use cases:
• Borrow copy of book
• Extend loan
Diagram:

Use case description:
1. Borrow a copy of the book: If the person wishes to borrow a book from Derpville Public Library, he/she must be a member of the library in which case they will be allowed to issue a certain number of books. If the person is not a member, the book will not be issued to them for taking away, rather they will have to sit and read in the library.
2. Extending loan: Some books will be lent for 3 weeks while others will be lent for more than 3 weeks in which case the person borrowing has to come to the library and get the date extended. There is a limit on how much the user can extend the date of a particular book.