L’Hôpital’s Rule

Calculus is a behavioral science. Granted, it doesn’t deal with the behaviors of humans. But it does make an enormous fuss about behaviors of functions. Functions do have their own behaviors and characteristics.

The Swiss mathematician Johann Bernoulli (I.) ...
Johann Bernoulli

One of the basic things we learn in calculus is the limiting process. Basically, it tells us what value the function is going nearest to, when your x is going towards a certain value. Usually, you get a numerical value for a limit, and sometimes you get an infinity, which just simply means your function is growing too big or too small, all too fast for your limits to know where your function’s going.

This is a good example on why you can’t always say that math is an exact science: some things are just too big, too small or too undefined to understand or compute for. And math isn’t just about computations, it’s about behavior.

But what if your limit is a 0/0 or an ∞/∞ ? What would that mean?

To find out, let’s take a page out of our good French Geometrician friend from the 1700’s Guillame de l’Hôpital’s book Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes. (Analyses of Curves Too Infinitely Small to Understand.) In this paper he published in 1696, he included a good number of lessons he learned from his Swiss teacher Johann Bernoulli (the first, not the second or the third) who, like a lot of his Bernoulli family members, was a well known genius in mathematics. Johann, specifically, was best when it came to discovering things about infinitesimal calculus.
So what did they tell us?

Definition: Indeterminate Form.

Let f(x) and g(x) be functions of x.

If  or ∞ and  , then  is of the indeterminate form 0/0 or ∞/∞.

Theorem: L’Hôpital’s Rule.

If  is of the indeterminate forms 0/0 or ∞/∞, then

So if you ever do encounter those two forms 0/0 or ∞/∞, then you know what to do. Just differentiate both the numerator and the denominator, and use the limiting process again. The rule also applies if we replace x→a with x→a+,x→a-,x→+∞ and x→-∞

Example 1.

If we evaluate the limit by plugging in the value x=2 in the equation, then we’d have:

Now, we see that evaluating the limit like any usual limit would give us an indeterminate form 0/0. We use L’Hôpital’s Rule. Let’s differentiate both the numerator and the denominator.

And then we evaluate just like usual.

And now that we have arrived at a known value for the limit, we have an answer.
Here are a couple more examples.

Example 2.

Example 3.

We see that after applying L’Hôpital’s Rule, we get the indeterminate form 0/0 again. It’s just a simple matter of repeating the process, or moving around the trigonometric identities to get an answer.

Don’t give up. Try using the trigonometric identities.

But there are more indeterminate forms.  Getting the answers is simply moving around the function to get an indeterminate form of 0/0 or ∞/∞ before you apply L’Hôpital’s Rule.

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