Today we're not going to solve problems. We are going to introduce the intuitive idea behind limits.

2. The Intuitive Idea of Limit
Let’s suppose we want to figure out the area of this circle:

3. The Intuitive Idea of Limit
Let’s suppose we want to figure out the area of this circle:
From basic geometry we know that its area is:

4. The Intuitive Idea of Limit
Let’s suppose we want to figure out the area of this circle:
From basic geometry we know that its area is:
A = πr2

5. The Intuitive Idea of Limit
Let’s suppose we want to figure out the area of this circle:
From basic geometry we know that its area is:
A = πr2
But let’s suppose we don’t know this magic formula!

6. The Intuitive Idea of Limit
Let’s say we have the following figure:

7. The Intuitive Idea of Limit
Let’s say we have the following figure:

8. The Intuitive Idea of Limit
Let’s say we have the following figure:

9. The Intuitive Idea of Limit
Let’s say we have the following figure:

10. The Intuitive Idea of Limit
Let’s say we have the following figure:

11. The Intuitive Idea of Limit
Let’s say we have the following figure:
This simple idea, called the method of exhaustion, was used by
Archimedes more than 2000 years ago.

12. Limits of Functions
Let’s consider a simple function:

13. Limits of Functions
Let’s consider a simple function:
f (x) = x2

14. Limits of Functions
Let’s consider a simple function:
f (x) = x2

15. Limits of Functions
Let’s consider a simple function:
f (x) = x2
When x approaches 1, f also approaches 1:

16. Limits of Functions
Let’s consider a simple function:
f (x) = x2
When x approaches 1, f also approaches 1:
lim
x→1
f (x) = 1

17. Limits of Functions

18. Limits of Functions
f (x) =
x2 if x = 1
0 if x = 1.

19. Limits of Functions
f (x) =
x2 if x = 1
0 if x = 1.
So, our function is a parabola with a hole at x = 1:

20. Limits of Functions

21. Limits of Functions
lim
x→1
f (x)?

22. Limits of Functions
lim
x→1
f (x) = 1 = f (1)

23. Limits of Functions
lim
x→1
f (x) = 1 = f (1)
This means that it doesn’t matter what is the value of f (1).