The Intuitive Idea of Limit
Let’s suppose we want to figure out the area of this circle:
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:
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
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!
The Intuitive Idea of Limit
Let’s say we have the following figure:
The Intuitive Idea of Limit
Let’s say we have the following figure:
The Intuitive Idea of Limit
Let’s say we have the following figure:
The Intuitive Idea of Limit
Let’s say we have the following figure:
The Intuitive Idea of Limit
Let’s say we have the following figure:
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.
Limits of Functions
Let’s consider a simple function:
Limits of Functions
Let’s consider a simple function:
f (x) = x2
Limits of Functions
Let’s consider a simple function:
f (x) = x2
Limits of Functions
Let’s consider a simple function:
f (x) = x2
When x approaches 1, f also approaches 1:
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
Limits of Functions
Limits of Functions
f (x) =
{x2 if x 6= 1
0 if x = 1.
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:
Limits of Functions
Limits of Functions
limx→1
f (x)?
Limits of Functions
limx→1
f (x) = 1 6= f (1)
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).