《高等数学英文1-3-10资料》由会员分享，可在线阅读，更多相关《高等数学英文1-3-10资料（21页珍藏版）》请在金锄头文库上搜索。
1、Chapter 1 Functions and Limits,1.3 The Limits of Functions,I. Intuitive Meaning of Limit,II. Rigorous Definition of Limit,III. One-sided limits,IV. The limit at Infinity,V. Properties of the Limit,Introduction of this section,Consider the function:,Q: What is happening to f (x) as x approaches 1 ?,I
2、. Intuitive Meaning of Limit,Consider the function:,Q: What is happening to f (x) as x approaches 1 ?,I. Intuitive Meaning of Limit,f (x) approaches 4 as x approaches 1.,In mathematical symbols, we write,I. Intuitive Meaning of Limit,means that when x is near but different from a, then f (x) is near
3、 A.,Q: But, what does near mean? How near is near?,f (x) approaches 4 as x approaches 1.,Consider the function:,I. Intuitive Meaning of Limit,Consider the function:,When,I. Intuitive Meaning of Limit,Consider the function:,When,We use the Greek letter to stand for arbitrary positive number,II. Rigor
4、ous Definition of Limit,Definition: To say that means that for each given (no matter how small), there is a corresponding such that , provided that .,when,a,A,y,x,o,Notes:,1. must be given first.,II. Rigorous Definition of Limit,shows that the limit of f (x) as x approaches a has nothing to do with
5、whether f (x) has meaning at x=a.,Geometric interpretation of,2. is to be produced and it will usually depend on .,Example 1,II. Rigorous Definition of Limit,Proof,Let be given.,So choose,then,implies that,If we want,we just need,Example 2,II. Rigorous Definition of Limit,Proof,Let be given.,Choose,
6、then,implies that,If we want,We just need,Example 3,II. Rigorous Definition of Limit,Proof,Let be given.,Choose,If we want,We first agree to make,We need,III. One-sided limits,Theorem 1,Example 4,III. One-sided limits,IV. The limit at Infinity,Consider the function,What happens to f (x) as x gets la
7、rger and larger?,f (x) gets smaller as x gets larger.,In mathematical symbols, we write,IV. The limit at Infinity,Consider the function,What happens to f (x) as x gets larger and larger?,M,-M,To say that means that for each given there is a corresponding number such that,Definition,Similarly,IV. The
8、 limit at Infinity,Example 5,Prove that ( k is a positive integer).,Proof,Let be given.,That is,If we need,We choose,IV. The limit at Infinity,Horizontal asymptote (水平渐近线),The line y = a is a horizontal asymptote of the graph of y = f (x) if either,IV. The limit at Infinity,V. Properties of the Limit,1. Uniqueness,Th:,2. Local boundedness,Th:,V. Properties of the Limit,3. Conservation of the sign,Corollary:,Th:,Th:,
