Learning Objectives

Recognize when to apply L’Hôpital’s rule.

Identify indeterminate forms produced by quotients, products, differences, and powers, and apply L’Hôpital’s rule in each case.

Describe the relative growth rates of functions.
In this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule, uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.
Applying L’Hôpital’s Rule
L’Hôpital’s rule can be used to evaluate limits involving the quotient of two functions. Consider
If then
However, what happens if and In this case, we have to deal with what’s called the indeterminate form of type This is considered an indeterminate form because we cannot determine the exact behavior of the quotient as approaches without further analysis. For example, consider the limits
The first limit can be evaluated by factoring the numerator:
As for , one can use a geometric argument to show that
Note that although both limits we just considered exist, in fact, anything is possible for the indeterminate form: the limit might exist and be equal to any real number , the limit might not exist, but there might be a trend of or , or there might be no limit and no trend. For example, knowing that , it is easy to show that
We are going to develop a universal technique for evaluating limits such as the ones above. Not only does it allow for an alternative and sometimes easier way to evaluate these limits, but also, and more importantly, it works for evaluating many other limits that we could not calculate before.
The idea behind L’Hôpital’s rule can be explained using local linear approximations. Consider two differentiable functions and such that and . For that is very close to we can write and (This follows either from the geometric understanding that the tangent line is very close to the graph around the point where the tangent “touches” the curve or, algebraically, from the formal limit definition of the derivative.)
Therefore,
Since is differentiable at it is continuous at and therefore Similarly, If we also assume that and are continuous at then and Using these arguments, we conclude that
Note that the assumptions that and are continuous at and can be loosened. We state L’Hôpital’s rule formally for the indeterminate form Also note that writing does not mean we are actually dividing zero by zero. It is just the notation we use when dealing with a limit of a quotient with both numerator and denominator approaching zero.
L’Hôpital’s Rule ( Case)
Suppose and are differentiable functions over an open interval containing except possibly at If and then
Proof
We provide a proof of this theorem in the special case when and are all continuous over an open interval containing In this case, since and and are continuous at we have that Therefore,
Note that L’Hôpital’s rule allows to calculate the limit of a quotient by considering the limit of the quotient of the derivatives It is important to realize that we are not calculating the derivative of the quotient
Applying L’Hôpital’s Rule ( Case)
Evaluate each of the following limits using L’Hôpital’s rule.
Solution

Since the numerator approaches and the denominator approaches 0 when , we can apply L’Hôpital’s rule to evaluate this limit. We have

As and Therefore, we can apply L’Hôpital’s rule. We obtain

When we have that , that is, the denominator approaches 0, and the numerator approaches . Therefore, we can apply L’Hôpital’s rule. We obtain

As both numerator and denominator approach zero: and . Therefore, we can apply L’Hôpital’s rule. We obtain
Since numerator and denominator of this new quotient both approach 0 as and , we apply L’Hôpital’s rule again. In doing so, we see that
Therefore, we conclude that
Evaluate
Answer
1
We can also use L’Hôpital’s rule to evaluate limits of quotients in which and Limits of this form are classified as indeterminate forms of type Again, note that we are not actually dividing by Since is not a real number, that is impossible; rather, is the notation we use when dealing with a limit of a quotient with both numerator and denominator having an infinite trend of or .
L’Hôpital’s Rule ( Case)
Suppose that and are differentiable functions over an open interval containing except possibly at Further suppose that and Then,
provided that the limit on the right exists or has a trend of or This result also holds for onesided limits or if the finite number is replaced with or
Applying L’Hôpital’s Rule ( Case)
Evaluate each of the following limits using L’Hôpital’s rule.
Solution

Since and are firstdegree polynomials with positive leading coefficients, and Therefore, we apply L’Hôpital’s rule and obtain
Note that this limit can also be calculated without invoking L’Hôpital’s rule, by dividing the numerator and denominator by the highest power of in the denominator:
L’Hôpital’s rule provides us with an alternative means of evaluating limits of this type.

Here, and Therefore, we can apply L’Hôpital’s rule and obtain
Now, as Therefore, the first term in the denominator is approaching zero and the second term is getting really large. In this case, anything can happen to the product, and we cannot make any conclusion yet. (Actually, this is an indeterminate form that will be discussed a bit later.) To evaluate the limit, we use the relation of to rewrite the quotient under the limit:
Now, and so we can apply L’Hôpital’s rule again:
We conclude that
Evaluate
Answer
0
As was already mentioned, L’Hôpital’s rule is an extremely useful tool for evaluating limits. It is important to remember, however, that to apply L’Hôpital’s rule to a quotient it is essential that the limit of be of the form or This will be illustrated in the example below.
When L’Hôpital’s Rule Does Not Apply
Consider Show that the limit cannot be evaluated by applying L’Hôpital’s rule.
Solution
Because the limits of the numerator and denominator are not both zero and are not both infinite, we cannot apply L’Hôpital’s rule. If we tried to do so, we would have erroneously concluded that
However, since and using the arithmetic properties of limits, we actually have that and so that
Another, trickier, case when L’Hôpital’s rule cannot be applied is when the limit of the quotient obtained after differentiation does not exist. Indeed, if we carefully read the statements of the L’Hôpital’s rule theorems, we can notice that they both say “provided that the limit on the right exists”. We explore the situation when this condition fails in the next example.
When L’Hôpital’s Rule Does Not Apply
Consider Show that the limit cannot be evaluated by applying L’Hôpital’s rule.
Solution
Both numerator and denominator approach infinity. Thus it appears that we can apply L’Hôpital’s rule. If we tried to do so, we would have erroneously concluded that
However, since neither exists, nor is infinite, L’Hôpital’s rule does not actually apply.
Note that this does not mean that the original limit fails to exist. Because , when , we have that
. Now since
(that can be proved using L’Hôpital’s rule), we can use the squeeze theorem to conclude that
Explain why we cannot apply L’Hôpital’s rule to evaluate Evaluate by other means.
Answer
Other Indeterminate Forms
We have seen that L’Hôpital’s rule is very useful for dealing with the indeterminate forms and . It can also help to evaluate limits involving other indeterminate forms such as , , , , and . As before, these expressions should be treated not as algebraic operations but as the notation reflecting the behavior of the function under the limit. We show why they are indeterminate forms and how to use L’Hôpital’s rule to evaluate the corresponding limits. The key idea is to rewrite the expression as a quotient of the form or .
Indeterminate Form of Type
Suppose we want to evaluate , where and as . Since one term in the product is approaching zero but the other term is becoming arbitrarily large in magnitude, anything can happen to the product. We use the notation to denote the form that arises in this situation. The expression is considered indeterminate because, without further analysis, we cannot determine the exact behavior of the product as .
For example, let n be a positive integer and consider . As , and . However, the limit of as varies, depending on . Indeed,
if , then ,
if , then ,
and if , then . (All these limits can be evaluated by either dividing both numerator and denominator by the highest power of in the denominator or by using L’Hôpital’s rule, multiple times, when needed).
We now consider another limit involving the indeterminate form and show how to rewrite the function as a quotient to use L’Hôpital’s rule.
Indeterminate Form of Type
Evaluate
Solution
To be able to use L’Hôpital’s rule, we rewrite the function as a quotient in the following way: . Since as and as , we can apply L’Hôpital’s rule to obtain
We conclude that
In general, one of the ways to transform a product of functions into a quotient is to use double reciprocals, that is, rewrite as or .
Evaluate
Answer
1
Indeterminate Form of Type
Suppose that (or ). Then the limit represents an indeterminate form .
To show that this is, indeed, an indeterminate form, and the answer could be anything, consider the following example. Let be a positive integer, and . As , and , but depends on the value of the exponent . To see this, we recall that, when , the behavior of a polynomial is determined by the leading term since and for any .
Therefore, if , then .
On the other hand, if , then .
Finally, if , then .
In the next example, we show how to rewrite an expression involving the indeterminate form as a fraction in order to apply L’Hôpital’s rule.
Indeterminate Form of Type
Evaluate
Solution
Bringing the expression to a common denominator, we obtain
Since and , we can apply L’Hôpital’s rule.
We have that and Because the denominator is positive as approaches zero from the right, we conclude that
Therefore,
In general, bringing the difference of fractions to a common denominator often works to reduce an indeterminate form to or form, after which L’Hôpital’s rule can be applied.
Evaluate
Answer
0
Hint
Rewrite the difference of fractions as a single fraction.
Another types of indeterminate forms that arise when evaluating limits involve exponents. The expressions and are all indeterminate forms. Again, these expressions are not meant to be evaluated using algebraic operations, rather they provide a notation used to describe the behavior of a function under the limit. We now demonstrate how L’Hôpital’s rule can be used to evaluate limits involving these indeterminate forms.
Suppose we want to evaluate and we arrive at one of the indeterminate forms listed above. We proceed as follows. Let Then, and we first evaluate . This allows to reduce an indeterminate power to an indeterminate product. Indeed, corresponds to , corresponds to , and corresponds to . We then use the techniques discussed earlier to rewrite the expression as a quotient so that we can apply L’Hôpital’s rule. Suppose that where may be or So we have that and since the natural logarithm function is continuous, we conclude that It follows that
Indeterminate Form of Type
Evaluate
Solution
Let Then
We first need to evaluate Since both numerator and denominator approach infinity as , we can apply L’Hôpital’s rule to obtain
So we have that and since the natural logarithm function is continuous, we conclude that
This leads to
Evaluate
Answer
Hint
Take and consider .
Indeterminate Form of Type
Evaluate
Solution
Let Then
We now evaluate Since and we have the indeterminate form To apply L’Hôpital’s rule, we need to rewrite as a fraction. We could write
or
Let’s consider the first option. In this case, applying L’Hôpital’s rule, we would obtain
Unfortunately, we not only have another expression involving the indeterminate form but the new limit is even more complicated to evaluate than the one we started with. Instead, we try the second option. By writing and applying L’Hôpital’s rule, we obtain
Using the fact that and we can rewrite the last limit as
Because , we can apply L’Hôpital’s rule again to get
We conclude that Using the continuity of logarithmic function, we obtain that , and hence
Evaluate
Answer
1
Hint
Take and consider .
Growth Rates of Functions
Suppose the functions and both approach infinity as Although the values of both functions become arbitrarily large as the values of become sufficiently large, sometimes one function is growing more quickly than the other. For example, and both approach infinity as However, as shown in the following table, the values of are growing much faster than the values of
10  100  1000  10,000  
100  10,000  1,000,000  100,000,000  
1000  1,000,000  1,000,000,000 
In fact,
As a result, we say is growing more rapidly than as On the other hand, for and although the values of are always greater than the values of for each value of is roughly three times the corresponding value of as as shown in the following table. In fact,
10  100  1000  10,000  
100  10,000  1,000,000  100,000,000  
341  30,401  3,004,001  300,040,001 
In this case, we say that and are growing at the same rate as
More generally, suppose and are two functions that approach infinity as We say grows more rapidly than as if
On the other hand, if there exist constants and a number such that , for all , we say that and grow at the same rate as .
Next we see how to use L’Hôpital’s rule to compare the growth rates of power, exponential, and logarithmic functions.
Comparing the Growth Rates of and
For each of the following pairs of functions, use L’Hôpital’s rule to evaluate
Solution

Since and we can use L’Hôpital’s rule to evaluate We obtain
Because and we can apply L’Hôpital’s rule again:
Hence, , which means that grows more rapidly than as (see the figure with the graphs of these functions below).
Growth rates of a power function and an exponential function. 5 10 15 20 25 100 225 400 148 22,026 3,269,017 485,165,195 
Since and we can use L’Hôpital’s rule to evaluate We obtain
Thus, grows more rapidly than as , which agrees with the following figure displaying the graphs of these functions.
Growth rates of a power function and a logarithmic function 10 100 1000 10,000 2.303 4.605 6.908 9.210 100 10,000 1,000,000 100,000,000
Compare the growth rates of and
Answer
The function grows faster than
Hint
Apply L’Hôpital’s rule sufficiently many times to evaluate
Using the same ideas as in example (a) above, it is not difficult to show that grows more rapidly than for any In the following table, we compare with and as becomes large.
5  10  15  20  
125  1000  3375  8000  
625  10,000  50,625  160,000  
148  22,026  3,269,017  485,165,195 
Similarly, it is also easy to show that grows more rapidly than for any In the table below, we compare with and when becomes large.
10  100  1000  10,000  
2.303  4.605  6.908  9.210  
2.154  4.642  10  21.544  
3.162  10  31.623  100 
Key Concepts

L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form or arises.

L’Hôpital’s rule can also be applied to other indeterminate forms if the expressions under the limit can be rewritten as a quotient of indeterminate form or

The exponential function grows faster than any power function

The logarithmic function grows slower than any power function
Exercises
For the following exercises, evaluate the given limit. If the limit does not exist, indicate whether there is a trend of or . In case you are using L’Hôpital’s rule, explain why it applies.
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(Hint: Factor out of the expression.)
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Glossary
 indeterminate forms
 when evaluating a limit, the forms and are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is
 L’Hôpital’s rule
 if and are differentiable functions over an open interval that contains except possibly at and or and are infinite, then provided the limit on the right exists or has a trend of or
Hint
Determine the limits of the numerator and denominator and analyze the behavior of the quotient.