Rational functions, Compute the limit, Substitute, Limit of the functions, Value of the function, Continuous, Factorize, 0/0, number/0, right side limit, left side limit.
How could we compute this limit?
The first step is to substitute into the function.
Let's see what we get.
If the result is interpretable, then we are finished.
The resulting number is the limit.
If the result is not interpretable,
then we are in trouble.
In this case there are usually two possibilities,
and sometimes there is a third one.
Let's see what should be done in the first two cases.
In this case we factorize both the numerator and the denominator.
In this case we factorize only the denominator.
In this case something will happen, too.
In other words, you have to factorize the part that is zero.
If both are zero, then both,
if only the denominator is zero, then only the denominator.
Let's see how.
Well, like this.
This certain here is the number that x is approaching.
If it happens that then it is 4.
Now we only have to figure out these.
For the quadratic case, there is a trick.
This happens to be quadratic, so let's check it out.
We have to ask ourselves some questions.
The first question is: what should we write here,
to get x2?
Well, x seems to be a good idea.
Everything is OK so far.
Now let's take a look at these.
No, we don't have to look at them.
They only confuse us, so let's make them a bit dimmer.
What we have to look at, is this.
And we have to figure out what we should multiply by negative 4 to get 20.
The same trick is used at the bottom, too.
Let's see one more:
Let's start with substituting 2.
If we get for example 42, then we are finished and we have nothing to do.
But we are not that lucky.
So, we have to resort to factorization, again.
Let's see how we can get 4x2.
Similar excitement can be expected at the bottom.
Now let's see these.
This other case will be somewhat more unpleasant.
In this case we factorize only the denominator,
and because of that, we cannot simplify.
But let's see a specific example.
We start again by substituting 2, because we may get lucky and get a specific number.
Luck is not on our side.
So, we factorize the denominator.
There is no point in factorizing the numerator,
and cannot be factorized anyway.
So that will stay.
We do the usual magic tricks on the denominator.
And now comes this part.
Here, we can substitute 2,
and this part will have some very funny things happening to it.
Let's take a look at this function.
If then .
But only from the left side.
Because if from the right side
This is very interesting, and the standard notation for this is:
In such cases, where the right limit and left limit are not the same, we say there is no limit.
There is one more thing.
We already learned in elementary school that we cannot divide by zero. So this one is not interpretable:
But these are.
If the denominator is approaching zero through the negative numbers, then the fraction is approaching negative infinity.
If the denominator is approaching zero through the positive numbers, then the fraction is approaching positive infinity.
The reason why this is important for us is that we can solve the previous problem without drawing.
We started drawing around here.
Now we will substitute instead of drawing.
This is not interpretable, but ...
We have to check from the left and right separately.
If then and are negative.
But if then and are positive.
The result is the same this way, too: there is no limit.