Essential Knowledge 1.1C3

Essential Knowledge 1.1C3 Students will know that limits of the indeterminate forms and may be evaluated using L'Hôpital's Rule.

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You can't learn L'Hôpital's Rule until you've learned derivatives, so don't try to come here straight from 1.1C2. You should only attempt to learn L'Hôpital's Rule once you have finished the entire "Big Idea 2: Derivatives"

So far we've learned some strategies for determining limits that are initially indeterminate. Now it's time to learn a beautiful approach created by L'Hôpital.

Let's look at an example

Using direct substitution, we get ∞/∞. None of the methods we've used thus far can change this into a determinable form. And yet, let's just think about this.

Remember, not all ∞s are the same. The ∞ in the numerator wants to make the answer ∞, the ∞ in the denominator wants to make the answer 0. They are having a tug of war, and the ∞ that increases faster will win. So let's just take a look at how the numerator and denominator are increasing.

Looking at this graph, we can see that e^x is increasing at a much greater rate than ln(x). Surely it will win the tug of war. In other words, the ∞ in the denominator will be much more powerful than the ∞ in the numerator.

Well when it comes to rates, there's only one way to get a precise answer: derivatives!

So if we want to compare who will win the tug of war, simply take the derivative of the numerator and the denominator separately. This is L'Hôpital's Rule, and it's stated like this.

For our example, let f(x) =ln(x) and let g(x) = e^x. Then f'(x) = 1/x and g'(x) = e^x and we get the following.

As predicted, the denominator won the tug of war, and the limit is 0.