《高等数学》课程电子教案（PPT课件）Chapter 1 Functions and Limits §1.3 The Limits of Functions

Chapter1LimitsFunctionsand$ 1.3 The Limits of Functions

Chapter 1 Functions and Limits §1.3 The Limits of Functions

Introduction of this sectionI Intuitive Meaning of LimitIl. Rigorous Definition of Limitll.One-sided limitsIV. The limit at InfinityV.Properties of theLimitS1.3 The Limitsof functions

§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

I.Intuitive Meaning ofLimit2x2 - 2Consider the function: f(x) =x-1Q: What is happening to f (x) as x approaches 1 ?0.93.8V1.14.240.993.981.014.020.9993.9980x1.0014.0020.9999993.9999981.0000014.000002S 1.3 The Limits of functions

§1.3 The Limits of functions Consider the function: , 1 2 2 ( ) 2 − − = x x f x Q: What is happening to f (x) as x approaches 1 ? I. Intuitive Meaning of Limit y o x 4 1 0.9 3.8 1.1 4.2 0.99 3.98 1.01 4.02 0.999 3.998 1.001 4.002 0.999999 3.999998 1.000001 4.000002

I.Intuitive Meaning ofLimit2x2 - 2Consider the function: f(x) =x-1Q: What is happening to f (x) as x approaches 1 ?VIn mathematical symbols, we write42x-2limAx-1x-10x-f (x)approaches 4asxapproaches 1.S 1.3 The Limits of functions

§1.3 The Limits of functions Consider the function: , 1 2 2 ( ) 2 − − = x x f x 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 4 1 2 2 lim 2 1 = − − → x x x y o x 4 1

I.Intuitive Meaning of Limit2x2 - 2Consider the function: f(x) =x-1f(x)approaches 4 as xapproaches 1.lim f(x) = A means that when x is nearbut different from a, then f(x) is near A.Q：But,what doesnear mean?How nearisnear?S1.3 The Limitsof functions

§1.3 The Limits of functions I. Intuitive Meaning of Limit means that when x is near but different from a, then f (x) is near A. f x A x a = → lim ( ) Q: But, what does near mean? How near is near? f (x) approaches 4 as x approaches 1. Consider the function: , 1 2 2 ( ) 2 − − = x x f x

I.Intuitive Meaning ofLimit2x2 - 2Consider the function: f(x) =x-1When x±1, f(x)-4=2x-1台x-1<0.052x-1<0.1 xS 1.3 The Limits of functions

§1.3 The Limits of functions I. Intuitive Meaning of Limit f (x) − 4 = 2 x −1  x −1  0.05 y o x 4 1 Consider the function: , 1 2 2 ( ) 2 − − = x x f x When x  1, 2 x −1  0.1

I.Intuitive Meaning of Limit2x2 -2Consider the function: f(x) =x-12When x±1, f(x)-4= 2x-12x-1<0.052x-1 <0.12|x -1| <0.01 |x -1 <0.005x12x-1 < 0.001 |x-1|<0.0005We usethe Greek letter to stand for arbitrarypositive numberD2x-1<8x-1<2S1.3 The Limitsof functions

§1.3 The Limits of functions I. Intuitive Meaning of Limit f (x) − 4 = 2 x −1  x −1  0.05 2 x −1   2 1   x −  y o x 4 1 Consider the function: , 1 2 2 ( ) 2 − − = x x f x When x  1, 2 x −1  0.1 2 x −1  0.01  x −1  0.005 2 x −1  0.001  x −1  0.0005 We use the Greek letter to stand for arbitrary positive number  2 x −1  

II.Rigorous Definition ofLimitDefinition:To say that lim f(x) = Ameans that for each given ε >0x→a(no matter how small), there is a corresponding > 0such that [f(x)- Al 0,3S>0,when 0<x-a<s,f(x)-A<8S 1.3 The Limits of functions

§1.3 The Limits of functions II. Rigorous Definition of Limit Definition: To say that means that for each given (no matter how small), there is a corresponding such that , provided that . f x A x a = → lim ( )   0   0 f (x) − A   0  x − a     0,   0, when 0  x − a   , f (x) − A   =  → f x A x a lim ( )

Il.RigorousDefinitionofLimitGeometric interpretation of lim f(x) = Ax-0y= f(x)yNotes:A+81.mustbegivenfirstA-82. S is to be produced and itwill usually depend on x0Lla-sa a+s3. 0<x-al<8 shows that the limit of f(x) as x approaches ahas nothingto do withwhetherf(x)has meaningatx=a.S1.3The Limitsoffunctions

§1.3 The Limits of functions A+  A−  a − a a +  A y o x y = f (x) Notes: 1. must be given first.  II. Rigorous Definition of Limit 3. shows that the limit of f (x) as x approaches a has nothing to do with whether f (x) has meaning at x=a. 0  x − a   f x A x a = → Geometric interpretation of lim ( ) 2. is to be produced and it will usually depend on .  

Il. Rigorous Definition of LimitExample1Prove that lim(3x -7) = 5.x→4ProofLet & be given. If we want[f(x)-A=|3x-7-5=3x-4 <8 ,Cwe just need38So chooseS3then0<|x-4|< implies that(3x -7)-5= 3x- 4<38 = 8.S 1.3 The Limits of functions

§1.3 The Limits of functions lim(3 7) 5. 4 − = → x x Example 1 Prove that II. Rigorous Definition of Limit Proof Let be given.  , 3  So choose  = then 0  x − 4   implies that . 3 4  x −  f (x) − A = 3x − 7 − 5 = 3 x − 4 If we want   , we just need (3x − 7) − 5 = 3 x − 4  3 = 