Welcome to Professor MAD.
Mathematics for Everyone.
Today we are going to talk about the basics of calculus, an introduction to limits.
Here we have a function Y.
Think we have to draw the plot for this.
Let's try to find the values of Y for different X values and draw the graph.
For X= -2 ,we can calculate Y= -1.
Let's put this value in the table and mark the point in our graph.
Next for X= -1, we can calculate Y=0.
Let's put this value in the table and mark the point in our graph.
Now we can start to draw the plot by connecting these two points.
Next for X=0, we can calculate Y=1.
Let's do the same.
Next for X= +1, we can calculate Y =? , oh ho we have a problem here.
For X=1, Y is equal to zero divided by zero.
What is the value of that?
Actually we can't calculate the value of a zero divided by zero.
So we say it's undefined.
Let's move on and try to find the value of X=2.
For X=2, Y=3.
But since we don't know the value of X=1, we can't continue the graph.
Let's try to find the values of Y Beyond this.
For X=3, Y is equal to 4.
The graph continues beyond that.
For X=4, Y is equal to 5.
Likewise we can find the Y for all the values for X, except X=1.
At X=1, we need another way to get an answer.
Even though we can't work this out directly,
we can see what it should be as we get closer and closer.
Instead of trying to work it out for X=1, let's try approaching it closer and closer.
There are two ways to approach X=1.
We can either approach X=1, from the values greater than the X=1,
or from the values less than X=1.
Let's first check with the values less than X=1.
For X= 0.5, Y=1.5.
For X= 0.6, Y=1.6.
For X= 0.7, Y= 1.7.
For X= 0.8, Y=1.8.
For X=0.9, Y=1.9.
Now check with the values greater than X= 1.
For X=1.5, Y= 2.5.
For X= 1.4, Y= 2.4.
For X=1.3, Y= 2.3.
For X= 1.2, Y= 2.2.
For X=1.1, Y= 2.1.
Now we see that as X gets close to 1, Y gets close to 2.
As a graph it looks like this.
In this graph there is a hole at X=1.
As we approach X=1 from both sides,
the value of y gets closer to 2.
So in truth we cannot calculate the value of y at X=1.
We are saying that as X approaches 1 from both sides,
the limit of Y is 2.
We also have a special notation to talk about limits.
This is how we would write the limit of Y as X approaches 1.
When the value of a function is undetermined for a specific X value,
we use the concept of the limit
The limit of a function describes the function's behavior
close to a particular X value.
It does not necessarily give the value of the function at X.
Limits are essential in calculus.
Especially when a function has undetermined values as in this example.
Thanks for watching.
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