Download - Day 1: Intuitive Idea and Notation
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The Intuitive Idea of Limit
Let’s suppose we want to figure out the area of this circle:
![Page 3: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/3.jpg)
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:
![Page 4: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/4.jpg)
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
![Page 5: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/5.jpg)
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!
![Page 6: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/6.jpg)
The Intuitive Idea of Limit
Let’s say we have the following figure:
![Page 7: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/7.jpg)
The Intuitive Idea of Limit
Let’s say we have the following figure:
![Page 8: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/8.jpg)
The Intuitive Idea of Limit
Let’s say we have the following figure:
![Page 9: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/9.jpg)
The Intuitive Idea of Limit
Let’s say we have the following figure:
![Page 10: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/10.jpg)
The Intuitive Idea of Limit
Let’s say we have the following figure:
![Page 11: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/11.jpg)
The Intuitive Idea of Limit
Let’s say we have the following figure:
This simple idea, called the method of exhaustion, was used byArchimedes more than 2000 years ago.
![Page 12: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/12.jpg)
Limits of Functions
Let’s consider a simple function:
![Page 13: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/13.jpg)
Limits of Functions
Let’s consider a simple function:
f (x) = x2
![Page 14: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/14.jpg)
Limits of Functions
Let’s consider a simple function:
f (x) = x2
![Page 15: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/15.jpg)
Limits of Functions
Let’s consider a simple function:
f (x) = x2
When x approaches 1, f also approaches 1:
![Page 16: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/16.jpg)
Limits of Functions
Let’s consider a simple function:
f (x) = x2
When x approaches 1, f also approaches 1:
limx→1
f (x) = 1
![Page 17: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/17.jpg)
Limits of Functions
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Limits of Functions
f (x) =
{x2 if x 6= 1
0 if x = 1.
![Page 19: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/19.jpg)
Limits of Functions
f (x) =
{x2 if x 6= 1
0 if x = 1.
So, our function is a parabola with a hole at x = 1:
![Page 20: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/20.jpg)
Limits of Functions
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Limits of Functions
limx→1
f (x)?
![Page 22: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/22.jpg)
Limits of Functions
limx→1
f (x) = 1 6= f (1)
![Page 23: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/23.jpg)
Limits of Functions
limx→1
f (x) = 1 6= f (1)
This means that it doesn’t matter what is the value of f (1).
![Page 24: Day 1: Intuitive Idea and Notation](https://reader033.vdocuments.site/reader033/viewer/2022060120/559288291a28ab56468b4746/html5/thumbnails/24.jpg)