What is L'Hopital's Rule?

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  • 0:06 Review of Limits
  • 1:57 L'H?pital's Rule Part 1
  • 5:11 L'H?pital's Rule Part 2
  • 6:28 Lesson Summary
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Lesson Transcript
Instructor: Robert Egan
A Swiss mathematician and a French mathematician walk into a bar ... and they walk out with the famous L'Hopital's rule for finding limits. In this lesson, learn what these two mathematicians came up with and how to use it to avoid the limit of zero divided by zero!

Limits Review

Mathmeticians LHopital and Bernoulli
Lhopital Picture

Let's take a minute to go back and think about limits. We've gotten really good at taking limits that look like this: limit as delta x goes to zero of (f(x + delta x) - f(x)) / delta x = f`(x), or the derivative of f(x). Even though we've gotten really good at taking these types of limits, we might have problems when it comes to a limit like this: limit as x goes zero of sin(x) / x. If we remember our limit rules, one thing we can do is divide and conquer. We can find the limit at the top of this equation, and we can find the limit at the bottom of this equation. Well, the limit of sin(x) as x goes to zero is zero. So far so good. And the limit of x as x approaches zero is zero.

So our limit is 0 / 0. Hmmm ... that's not so good. What do we do when we have limits like this? What do we do when our limit is equal to 0 / 0 or infinity / infinity? How do we make sense of that? Is there an actual number that this limit is; is it going to infinity or zero? How do we know? Well, I can graph this sin(x) / x, and it looks like the limit is 1. But how would I find that without graphing? How would I find that for more complicated equations that are harder to graph? How would I do this without my calculator?

L'Hopital's Rule Part One

The formal explanation of the rule
Lhopitals Rule Explanation

Well, two men are credited with solving this problem in the 17th century: the French mathematician Guilliame Francois Antoine de L'Hopital and the Swiss mathematician Johann Bernoulli. The story is borderline scandalous, since it's believed that Bernoulli came up with the solution and L'Hopital published it, so the solution is known as L'Hopital's rule.

So what's the essence of L'Hôpital's rule? Well, if you have two functions that are approaching zero, then we can't look at the value of the functions at those points to determine the limit of the ratio. However, we can look at the derivative. Maybe one of those functions is approaching zero faster than the other. Instead of finding the limit sin(x) / x, let's look at the limit of the derivatives. Maybe, by looking at the derivatives, we can get an idea of what happens to sin(x) / x at x=0.

Formally, L'Hopital's rule says that if you have some functions, like f(x) and g(x), and both of them approach zero as x goes to some number, like C, then the limit as x approaches C of the ratio of these functions is equal to the limit as x approaches C of the ratio of the derivatives of these functions. Thus, the limit as x goes to C of f(x) / g(x) is equal to the limit as x goes to C of f`(x) / g`(x).

Using the rule in the first example problem
Lhopital Example 1

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