Advanced Mathematics

1. Advanced Mathematics Teacher: Liubiyu

2. Chapter 1 Function and its graph Chapter 2 Limits and Continuity Chapter 1-2

3. Contents §1.1 Sets and the real number §1.2 Elementary functions and graph §2.1 Limits of Sequence of number §2.2 Limits of functions §2.3 The operation of limits §2.4 The principle for existence of limits §2.5 Two important limits §2.6 Continuity of functions §2.7 Infinitesimal and infinity quantity, the order for infinitesimals

4. §2.5 Two important limits D B 1 x C o A The following two important limits should be memorized. They will often be used to find limits of other more complicated functions. We now prove it precisely. Proof

5. D B 1 x C o A Area of the triangle OBC < Area of the sector OBC < Area of the triangle OCD (1) Now, by the formula for these area. So inequalities (1) take the form

7. Example 1 Solution Example 2 Solution

8. Example 3 Solution

9. Example 4 Solution

10. Example 5 Solution

11. Proof

16. ！ Notations

17. Example 7 Solution

18. Example 8 Solution Example 9 Solution