Notes, Lesson 4.5
Indeterminate Forms and L'Hopital's Rule

the title of this lesson should really be:

Indeterminate Forms and Bernoulli's Rule

L'Hopital's Rule was first published by the French nobleman, the Marquis de L'Hopital in the first ever published calculus textbook, "Analyse des infiniment petits pour l'intelligence des lignes courbes" in 1692. But the famous rule which was published in this book was taught to L'Hopital by his teacher Johann Bernoulli in 1691. Bernoulli is the discoverer of the rule. In the introduction to his textbook L'Hopital acknowledges his indebtedness to Leibniz, Jacob Bernoulli and Johann Bernoulli but L'Hôpital regarded the foundations provided by him as his own ideas. Here is part of that introduction:

 "I must own myself very much obliged to the labours of Messieurs Bernoulli, but particularly to those of the present Professor at Groeningen [Johann Bernoulli], as having made free with their Discoveries as well as those of Mr. Leibnitz: So that whatever they please to claim as their own I frankly return them. I must here in justice own (as Mr. Leibnitz himself has done, in Journal des Scavans for August, 1694) that the learned Sir Isaac Newton likewise discovered something like the Calculus Differentialis, as appears by his excellent Principia, published first in the Year 1687 which almost wholly depends on the Use of the said Calculus. But the Method of Mr. Leibnitz's is much more easy and expeditious, on account of the Notation he uses, not to mention the wonderful assistance it affords on many occasions."  Johann Bernoulli 1667-1748, Basel Switzerland Marquis de L'Hôpital 1661-1704, Paris France

L'Hopital's Rule and What it Empowers us to Do

Up to this point in our course, we have not been able to find the limits of all types of expressions. There are several indeterminate forms which have caused us to attempt to find algebraic simplifications for. Without these simplifications, we were not able to calculate these limits. L'Hopital's Rule allows us to now calculate limits for rational expressions with four different indeterminate forms. Here is L'Hopital's Rule:

When we consider the limit: , we can run into several different situations. These are shown in the table below:

 Condition Calculation of Limit Cannot be calculated. Of course division by zero is never possible.The limit may or may not exist. Cannot be calculated. Here the numerator and denominator are engaged in a war. If the numerator wins, then the limit could go to positive or negative infinity. If the denominator wins, then the limit could go to zero. We cannot determine the limit. We cannot tell if the limit exists or doesn't exist. Cannot be calculated. Again, division by zero is never possible.The limit may or may not exist. This of course can be calculated and L'Hopital's Rule

L'Hopital's Rule applies to the first two of the above situations. Here is L'Hopital's Rule:

 If you have either of the two interminate forms: or , then if the limit on the right side exists or is .

Here are some examples to illustrate the rule.

Example 1

 Find Given Problem. (Problem #8 in text) Notice that the inderminate form that we have is the form , so L'Hopital's rule can be used. We apply L'Hopital's rule, and then we can evaluate the limit.

Example 2: In this example we see that L'Hopital's rule may be applied repeatedly.

 Find . Given Problem. (Problem #16 in text) Notice that the indeterminate form that we have is the form , so L'Hopital's rule can be used. Here we apply L'Hopital's rule and differentiate the numerator and denominator seperately. But notice that this still leads us to the indeterminate form . We apply L'Hopital's rule a second time and this time, after seperately differentiating both numerator and denominator, we come up with a limit that we can calculate. Success.

Indeterminate Products

Another limit problem that we may want to be able to calculate is when and . Here again there is a war between zero and infinity as they are being multiplied by each other. We find that with a little creativity, we can use L'Hopital's rule here also. We can think of as being: . And since we can see that we have the indeterminate form of and L'Hopital's rule can be used. And so the "new" indetermindate form that L'Hopital's Rule can be used on is: .

Example 3:

 Find Given Problem. (Problem #22 in text). We do some algebra to get the limit into a form where L'Hopital might be applied. is of the form We now take a very important step and check if this limit is in one of the indeterminate forms that L'Hopital's Rule must have to be applied. And we see that it is of the form . Now we apply L'Hopital's Rule 3 times.

Indeterminate Differences

Another limit problem that we may want to be able to calculate is when and . We have here yet another war between the two functions. Using algebra and substitutions, this too may be able to be converted into the quotient form that gives us one of the two forms or .

Example 4:

 Find Given Problem. (Problem #28 in text). We first use some trigonometric identities to make these substitutions. Here we use the common denominator to put the expression into 1 ratio. It is of the form Now we check if it is in one of the forms which L'Hopital applies to. Now we applie L'Hopital's Rule once, and we have successfully calculated the limit!

L'Hopital's Rule @ Platonic Realms Check Concepts #1: True or False: L'Hopital discovered L'Hopital's Rule. Choose One  True  False #2: True or False: L'Hopital wrote the first calculus textbook. Choose One  True  False #3: True or False: L'Hopital's Rule helps you to calculate derivatives that could otherwise not be calculated. Choose One  True  False #4: L'Hopital's Rule may be applied ________ time(s) if you have either of the indeterminate forms or . Choose One  1  2  3  any number #5 True or False: L'Hopital's Rule applies to all indeterminate forms. Choose One  True  False