The numerator stays, and some tricks will be used again for the denominator.

Then we separate them.

We can substitute into this part.

The left and right side limits are not the same, so there is no limit.

In such number/zero cases this is what happens usually.

But not always.

If the denominator is something to the second power, then that is positive for sure.

Then there is a limit, and it is positive infinity.

Or negative infinity.

If the denominator is something cubed, then it could be either this or that.

If the exponent is even, then there is a limit.

If the exponent is odd, then there isn't.

Here we have a few more exciting cases.

Let's see what we get if we substitute 4.

Well, this means we should factorize both the numerator and the denominator.

We factor out x in the numerator and then it is finished.

But unfortunately, the radical in the denominator causes some minor problems.

Somehow, we should make magically appear there, too, but we need a trick for that.

We will rationalize the denominator.

Here are a few cases with some higher degree expressions in the limits.

In such cases it is worth to try factoring out.

In most cases it is successful. Sometimes it is not: in those cases people with stronger nerves can try polynomial division, and people with weaker nerves can start panicking.

Let's see what we could factor out.

Then we factorize the denominator.

And then we simplify.

First we factor out here, too,

and then we factorize.

However, minor excitements happen when we try to simplify.

We may need a few new formulas, too. Well, here they are:

Let's see how we can use them.

Now we will take a look at a really funny case.

We substitute 3,

and then we see it is 0/0 type.

So, we have to factorize both the numerator and the denominator.

On the top we can factor out x3.

Only there is a little trouble: both the numerator and the denominator are of the fourth degree.

We could factor out x3 on the top,

but to figure out THINGY, we will need a horrific polynomial division.

To figure out SOMETHING and THINGY, we will need a horrific polynomial division.

Function is continuous at , if it is interpretable at , and it has a limit that is finite at , and most importantly:

Let's see an example.

Is the following function continuous at 3?

Well, those who have the ability to see things, know it right away that it isn't,

because this function jumps at 3, and jumping is very bad for a function's continuity.

Let’s see how we could get this without drawing.

We compute the limit,

and then the value of the function,

and if they are equal, then the function is continuous, if they are not, then it is not continuous.

They are not equal, so the function is not continuous at 3.

Not continuous at 3, but it could be made continuous.

All we have to do is modify this.

And now it is continuous.

It is not that simple at 4.

It would be rather difficult to make this function continuous.

In fact, impossible.