Sec 2.4: The Precise Definition of a Limit Two positive small number (the Greek letter delta) (the Greek letter epsilon)

Sec 2.4: The Precise Definition of a Limit It is an interval centered at 2 with radius 1.5 It is an interval centered at 2 with radius 1.5 without the center {2}

Sec 2.4: The Precise Definition of a Limit It is an interval centered at 3 with radius 0.5 It is an interval centered at 3 with radius 0.5 without the center {3}

Sec 2.4: The Precise Definition of a Limit delta-interval It is an interval centered at with radius It is an interval centered at with radius without the center

Sec 2.4: The Precise Definition of a Limit It is an interval along the y-axis centered at 2 with radius 1.5

Sec 2.4: The Precise Definition of a Limit It is an interval along the y-axis centered at 2 with radius 1.5

Sec 2.4: The Precise Definition of a Limit epsilon-interval It is an interval centered at with radius

Sec 2.4: The Precise Definition of a Limit epsilon-interval It is an interval centered at with radius delta-interval It is an interval centered at with radius

Sec 2.4: The Precise Definition of a Limit We say the image of 4 is 2 What is the image of 6

Sec 2.4: The Precise Definition of a Limit What is the image of 4 What is the image of the interval (3, 5)

Sec 2.4: The Precise Definition of a Limit What is the image of the interval (10/7, 10/3)

Sec 2.4: The Precise Definition of a Limit We say the image of 4 is 2 We say the pre-image of 2 is 4 What is the pre-image of -1

Sec 2.4: The Precise Definition of a Limit What is the pre-image of the interval (4, 10)

Sec 2.4: The Precise Definition of a Limit What is the pre-image of the interval (0.4, 0.6)

Sec 2.4: The Precise Definition of a Limit What is the pre-image of the epsilon-interval

Definition: Sec 2.4: The Precise Definition of a Limit for every positive number there is a positive number means that such that (*) For every epsilon-interval around L there is a delta-interval around a such that The image of this delta-interval contained inside the epsilon-interval

Definition: Sec 2.4: The Precise Definition of a Limit for every positive number there is a positive number means that such that (*)

Definition: Sec 2.4: The Precise Definition of a Limit for every positive number there is a positive number means that such that (*) NOTE: Give me any epsilon-interval, I will give you a delta-interval satisfying condition (*)

Sec 2.4: The Precise Definition of a Limit for every positive number there is a positive number means that such that (*) Two types of problem 1) Given epsilon find delta 2) Prove graph equation

Sec 2.4: The Precise Definition of a Limit 1) Given epsilon find delta (graph) Steps: 1) Identify a, L, f(x), epsilon 2) Draw horizontal lines 3) Find the intersection points 4) Make sure that 5) Find the two deltas 6) Now choose your delta to be

Sec 2.4: The Precise Definition of a Limit 1) Given epsilon find delta (graph)

Sec 2.4: The Precise Definition of a Limit

Sec 2.4: The Precise Definition of a Limit 1) Given epsilon find delta (equation)   Steps: 1) Identify a, L, f(x), epsilon 2) Solve these equations 3) Make sure that 5) Find the two deltas 6) Now choose your delta to be

Sec 2.4: The Precise Definition of a Limit for every positive number there is a positive number means that such that (*) Two types of problem 1) Given epsilon find delta 2) Prove graph equation

Sec 2.4: The Precise Definition of a Limit 2) Prove SOLUTION identify 2) guessing a value for 3) Proof (showing that this works)

Sec 2.4: The Precise Definition of a Limit 2) Prove SOLUTION 1) Identify: 2) guessing a value for delta This suggests that we should choose 3) Proof (showing that this works) Thus Therefore, by the definition of a limit

Sec 2.4: The Precise Definition of a Limit Example: Use the graph of to find a number such that

Sec 2.4: The Precise Definition of a Limit 3.7321 4.2361

Sec 2.4: The Precise Definition of a Limit there is a positive number for every positive number such that means that Part-1 Part-2 Part-3

Precise Definition of an infinite Limit   Definition: there is a positive number for every positive number M such that means that