L’Hôpital’s Rule and Indeterminate Forms
October 9, 2026

[00:00:05] Okay, we tried direct substitution. We get one squared minus one. One squared plus one minus two.
[00:00:20] And we get zero over zero. What is zero over zero? An indeterminate form. Okay, so how would we
[00:00:28] solve this one? Oh, before we learn about L'Hôpital's rule, what's the only way we know how to solve this?
[00:00:45] Factor. Okay, we factor. X minus one, X plus one. X plus two, is that right? Yeah. Okay, now because X is never one in a limit, it's only approaching one.
[00:01:12] Now we plug it in: one plus one over one plus two. Two thirds. We can use L’Hôpital’s Rule for two specific indeterminate forms: zero over zero or infinity over infinity.
[00:01:38] Okay, and here is how L'Hôpital's rule works. If we get one of these two forms, the limit as X approaches any a,
[00:02:05] so it turns out it's the limit.
[00:02:21] We take the derivative of the numerator and the derivative of the denominator.
[00:02:35] Separately, not the derivative of the whole function, derivative of the numerator divided by the derivative of the denominator. Then we find the limit.
[00:02:48] Okay, so this is L'Hôpital's rule.
[00:02:51] So let's go back to this problem.
[00:03:03] We know it is an indeterminate form. You do need to check that: zero over zero or infinity over infinity.
[00:03:16] I always put L’Hôpital’s Rule above the equal sign. This is the limit as x approaches one. What is the derivative of x squared minus one?
[00:03:32] Two X. Two X. The derivative of this is two X plus one. Okay. And now we try solve this. Can we use direct substitution? We can. At least we try it. This is definitely, well, this is a rational function.
[00:04:01] If we get a number, we know it is continuous there, so we use direct substitution. We get two times one over two times one plus one: two thirds.
[00:04:13] This will help us solve lots of limits very easily compared with our old methods. Why didn’t I teach this in Unit 1? We didn’t know derivatives.
[00:04:32] There was no way to find a derivative before you knew what a derivative was. What is a derivative? It is a limit of a difference quotient.
[00:04:50] Okay. We didn't really do ones like this before. But now we actually can solve these.
[00:05:03] As x approaches infinity, x cubed tends to infinity; subtracting three still tends to infinity. E to the x also tends to infinity. This is an indeterminate form where we can use L’Hôpital’s Rule.
[00:05:30] Use L’Hôpital’s Rule. The derivative of x cubed minus three is three x squared. The derivative of e to the x is e to the x. Try again: both numerator and denominator tend to infinity.
[00:05:56] Now what? Do it again: L’Hôpital’s Rule. The derivative of three x squared is six x. The derivative of e to the x is e to the x. We still get infinity over infinity one more time.
[00:06:32] Six over e to the x. As x approaches infinity, the denominator grows without bound. Where does that go? It goes to zero. A fixed number over a denominator tending to infinity tends to zero.
[00:06:47] So we know the answer to this. It's one of the ones we memorized and the answer is? One.
[00:06:55] One. If you try direct substitution, sine of zero is zero. The denominator is zero, so the form is zero over zero. We can use L’Hôpital’s Rule here.
[00:07:18] Derivative of sine of X is? Cosine of X. Derivative of X is? One. Okay. Now we use direct substitution. Cosine of zero is? One. One. Beautiful.
[00:07:36] Okay. Go to the backside example.
[00:07:39] Three.
[00:08:03] Three.
[00:08:19] Okay.
[00:08:20] Okay.
[00:08:27] So if we try direct substitution, what is F of three?
[00:08:35] It's right here. This is three zero.
[00:08:40] And we plug in three here. We get zero.
[00:08:46] Zero over zero.
[00:08:51] Okay.
[00:08:56] So what are we going to use?
[00:09:03] Okay.
[00:09:10] Which is going to be F prime of X over two X.
[00:09:18] The denominator is easy.
[00:09:23] Okay.
[00:09:24] What is F prime of three?
[00:09:31] This is the slope of the tangent line.
[00:09:40] Of F at X equals three, right?
[00:09:50] So what is the slope of the tangent line?
[00:09:54] It's the slope of the line.
[00:09:57] And what is the slope of that line?
[00:10:00] Negative two.
[00:10:02] We get negative one-third. Beautiful.
[00:10:08] Is there another way to solve this without L’Hôpital’s Rule?
[00:10:18] We actually could.
[00:10:22] Okay. F of X is what kind of a function?
[00:10:33] Oh, is it a linear function?
[00:10:36] That doesn't look like a linear function.
[00:10:39] A piecewise function.
[00:10:42] Each piece is linear.
[00:10:45] Okay.
[00:10:46] If all we care about is three, all we care about is this piece.
[00:10:49] Could we come up with the equation of that line?
[00:10:52] Yeah. What is the equation of that line?
[00:10:57] We know the slope, which is?
[00:11:01] Negative two.
[00:11:05] And then, yeah, plus B.
[00:11:09] And then, where's that? Three zero?
[00:11:11] Uh-oh.
[00:11:13] Okay. You could come up with the equation and then plug that in and then factor.