Limit of the functions, Value of the function, Jump, Continuous, Factorize, Find the limits.
Let's talk about limits.
Here is a harmless function
that we make a bit more exciting using these conditions.
When the function was the boring , it took 3, and assigned 6 to it.
But now, after the change, it assigns 8.
We write it like this
and we say that the function at 3 takes on the value of 8.
This is called the value of the function.
At the same time, as x approaches 3,
the values of the function approach 6.
From the other side, too.
The statement that "if then " is read "the limit of the function as x approaches 3 is 6", and it is written this way:
Let's see what happens if – for instance – .
The value of the function
To find the limit, we have to ask ourselves the question: if ,
then what does approach?
Well, it seems that
Therefore the limit
So, there are some x values in the function’s life where the limit and the value of the function are not the same, and there are some x values where those are the same.
For our function, there is only one x where the limit and the value of the function are different.
This is x=3.
This minor inconvenience happens to be at x = 3.
Here, the function makes a jump; everywhere else it behaves normally.
This normal behavior is called: "the function is continuous".
We call the function continuous at x where the limit and the value of the function are the same.
This will be our method to identify continuity:
We will compute the limit, then the value of the function, and then we ask ourselves whether those two resulting numbers are the same.
Let's see a few limits.
Here is this function, for instance.
Let's see what this limit is:
Well, this is a continuous function if substituting 2 results in
It takes a similarly large effort to compute this one:
But before we get too confident, let's see this one:
If we substitute 2 in the place of x, we get
And this presents certain problems.
If you don't believe it, type it into your calculator and see.
Luckily, there is a trick.
We factorize the numerator:
Then we simplify.
Now we can substitute 2.
So, if we have a rather unpleasant case, like this one:
then the most important thing is to remain calm,
and then try to factorize it.
Then we simplify,
and then we can substitute.
The next slideshow will reveal that this whole thing is simpler than we ever thought.