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