1. Teacher: Liubiyu

2. Chapter 1-2

3. Contents §1.2 Elementary functions and graph §2.1 Limits of Sequence of number §2.2 Limits of functions §1.1 Sets and the real number §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. New Words 绝对值 Absolute value 绝对值 几何意义 The geometric significance 几何意义 单边极限 One-Sided Limits 单边极限 右极限 Right limit 右极限 左极限 Left limit 左极限 使得 Such that 使得

5. §2.2 Limits of functions This section will extend the concepts and results obtained from sequences to function. As we know, a sequence is a special function f (n) defined on a number set N, its independent variable n is a discrete variable and its limit can only be infinite. However, for a function f (x), its independent variable x changes continuously and may have various limits. This makes the situation more complex. 1 、 Limit of a function for x tending to infinity Definition 1 Definition 1 ( informal definition ) Given a real number A, if the values f (x) of a function

6. approaches or equal A as the value of x approaches infinity, then f (x) has a limit A as x approaches .. Now, the question is how to describe this phenomenon precisely in mathematical language.

7. Therefore, we have the following definition. Definition 2 Definition 2 ( formal definition )

8. The geometric meaning of the definition above is asfollows: Notations (1) Similarly, we can define:

10. Example 1 Proof

11. Example 2 Proof

12. 2 、 Limit of a function for x tending to a finite value x 0

13. Therefore, we have the following definition: Definition 3

14. The geometric meaning of the definition is as follows:

15. Notations Example 3 Proof

16. Example 4 Proof

17. Example 5 Proof

18. Example 6 Proof

19. Example 7 Proof

20. 3 、 One-sided limits The right limit The left limit

21. Notations Example 8

22. Solution Notations

23. 4 、 The properties of functional limits Theorem 1 Theorem 1 ( uniqueness ) Theorem 2 Theorem 2 ( local boundedness )

24. Proof Theorem 3 Theorem 3 ( local preservation of sign )

25. Theorem 4 ) Theorem 4 ( local monotone property ) All these conclusions can be proved by definition of limit.