Let's see a few limits.

Now we will use the magic tricks learned in the previous slideshow.

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.