Now we will revolutionize the way we compute function limits.

More precisely, we will develop a very useful method for types of and .

The trick is that instead of the quotient of the original functions f and g, we look at the quotient of their derivatives.

We can do this, because those two limits are the same.

At least for limits of and type, if a few conditions are met.

Let’s see a few examples.

Now we can use l’Hopital’s Rule.

We may have to use l’Hopital’s Rule twice in a row.

Sometimes we have to use it more than twice.

And then sometimes we need to know when to stop.

If we used l’Hopital’s Rule again here, the derivative of the numerator would become quite ugly.

So, instead, we divide by x.

And now comes the real thrill.

L’Hopital’s Rule works for type cases as well. All we have to do is turn the product into a fraction.

Let's see how.

Here we have a few limits that may be useful for the rest of our lives:

A FEW IMPORTANT LIMITS

And here we have a few limits that would be quite difficult to compute using other methods.

Here we could try another round of l’Hopital’s Rule, but that would turn things rough. So instead, we exercise self-control and divide by .