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

2. 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}

3. 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}

4. 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

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

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

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

8. 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

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

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

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

12. 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

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

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

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

16. 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

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

18. 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 (*)

19. 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

20. 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

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

22. Sec 2.4: The Precise Definition of a Limit

23. 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

24. 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

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

26. 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

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

28. Sec 2.4: The Precise Definition of a Limit
3.7321
4.2361

29. 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

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