L'Hôpital's rule


In mathematics, more specifically calculus, L'Hôpital's rule or L'Hospital's rule provides a technique to evaluate limits of indeterminate forms. Application of the rule often converts an indeterminate form to an expression that can be easily evaluated by substitution. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the rule is often attributed to L'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician Johann Bernoulli.
L'Hôpital's rule states that for functions and which are differentiable on an open interval except possibly at a point contained in, if for all in with, and exists, then
The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be evaluated directly.

History

published this rule in his 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes, the first textbook on differential calculus. However, it is believed that the rule was discovered by the Swiss mathematician Johann Bernoulli.

General form

The general form of L'Hôpital's rule covers many cases. Let and be extended real numbers. Let be an open interval containing or an open interval with endpoint . The real valued functions and are assumed to be differentiable on except possibly at, and additionally on except possibly at. It is also assumed that Thus the rule applies to situations in which the ratio of the derivatives has a finite or infinite limit, but not to situations in which that ratio fluctuates permanently as gets closer and closer to.
If either
or
then
Although we have written x → c throughout, the limits may also be one-sided limits, when is a finite endpoint of.
In the second case, the hypothesis that diverges to infinity is not used in the proof ; thus, while the conditions of the rule are normally stated as above, the second sufficient condition for the rule's procedure to be valid can be more briefly stated as
The hypothesis that appears most commonly in the literature, but some authors sidestep this hypothesis by adding other hypotheses elsewhere. One method is to define the limit of a function with the additional requirement that the limiting function is defined everywhere on the relevant interval except possibly at. Another method is to require that both and be differentiable everywhere on an interval containing.

Requirement that the limit exist

The requirement that the limit
exists is essential. Without this condition, or may exhibit undamped oscillations as approaches, in which case L'Hôpital's rule does not apply. For example, if, and, then
this expression does not approach a limit as goes to, since the cosine function oscillates between and. But working with the original functions, can be shown to exist:
In a case such as this, all that can be concluded is that
so that if the limit of f/g exists, then it must lie between the inferior and superior limits of f′/g′.

Examples

Sometimes L'Hôpital's rule does not lead to an answer in a finite number of steps unless some additional steps are applied. Examples include the following:
A common pitfall is using L'Hôpital's rule with some circular reasoning to compute a derivative via a difference quotient. For example, consider the task of proving the derivative formula for powers of x:
Applying L'Hôpital's rule and finding the derivatives with respect to of the numerator and the denominator yields
as expected. However, differentiating the numerator required the use of the very fact that is being proven. This is an example of begging the question, since one may not assume the fact to be proven during the course of the proof.

Counterexamples when the derivative of the denominator is zero

The necessity of the condition that near can be seen by the following counterexample due to Otto Stolz. Let and Then there is no limit for as However,
which tends to 0 as. Further examples of this type were found by Ralph P. Boas Jr.

Other indeterminate forms

Other indeterminate forms, such as,,,, and, can sometimes be evaluated using L'Hôpital's rule. For example, to evaluate a limit involving, convert the difference of two functions to a quotient:
where L'Hôpital's rule is applied when going from to and again when going from to.
L'Hôpital's rule can be used on indeterminate forms involving exponents by using logarithms to "move the exponent down". Here is an example involving the indeterminate form :
It is valid to move the limit inside the exponential function because the exponential function is continuous. Now the exponent has been "moved down". The limit is of the indeterminate form, but as shown in an example above, l'Hôpital's rule may be used to determine that
Thus

Stolz–Cesàro theorem

The Stolz–Cesàro theorem is a similar result involving limits of sequences, but it uses finite difference operators rather than derivatives.

Geometric interpretation

Consider the curve in the plane whose -coordinate is given by and whose -coordinate is given by, with both functions continuous, i.e., the locus of points of the form. Suppose. The limit of the ratio as is the slope of the tangent to the curve at the point. The tangent to the curve at the point is given by. L'Hôpital's rule then states that the slope of the curve when is the limit of the slope of the tangent to the curve as the curve approaches the origin, provided that this is defined.

Proof of L'Hôpital's rule

Special case

The proof of L'Hôpital's rule is simple in the case where and are continuously differentiable at the point and where a finite limit is found after the first round of differentiation. It is not a proof of the general L'Hôpital's rule because it is stricter in its definition, requiring both differentiability and that c be a real number. Since many common functions have continuous derivatives, it is a special case worthy of attention.
Suppose that and are continuously differentiable at a real number, that, and that. Then
This follows from the difference-quotient definition of the derivative. The last equality follows from the continuity of the derivatives at. The limit in the conclusion is not indeterminate because.
The proof of a more general version of L'Hôpital's rule is given below.

General proof

The following proof is due to, where a unified proof for the and indeterminate forms is given. Taylor notes that different proofs may be found in and.
Let f and g be functions satisfying the hypotheses in the General form section. Let be the open interval in the hypothesis with endpoint c. Considering that on this interval and g is continuous, can be chosen smaller so that g is nonzero on.
For each x in the interval, define and as ranges over all values between x and c.
From the differentiability of f and g on, Cauchy's mean value theorem ensures that for any two distinct points x and y in there exists a between x and y such that. Consequently, for all choices of distinct x and y in the interval. The value g-g is always nonzero for distinct x and y in the interval, for if it was not, the mean value theorem would imply the existence of a p between x and y such that g' =0.
The definition of m and M will result in an extended real number, and so it is possible for them to take on the values ±∞. In the following two cases, m and M will establish bounds on the ratio.
Case 1:
For any x in the interval, and point y between x and c,
and therefore as y approaches c, and become zero, and so
Case 2:
For every x in the interval, define. For every point y between x and c,
As y approaches c, both and become zero, and therefore
The limit superior and limit inferior are necessary since the existence of the limit of has not yet been established.
It is also the case that
and
In case 1, the squeeze theorem establishes that exists and is equal to L. In the case 2, and the squeeze theorem again asserts that, and so the limit exists and is equal to L. This is the result that was to be proven.
In case 2 the assumption that f diverges to infinity was not used within the proof. This means that if |g| diverges to infinity as x approaches c and both f and g satisfy the hypotheses of L'Hôpital's rule, then no additional assumption is needed about the limit of f: It could even be the case that the limit of f does not exist. In this case, L'Hopital's theorem is actually a consequence of Cesàro–Stolz.
In the case when |g| diverges to infinity as x approaches c and f converges to a finite limit at c, then L'Hôpital's rule would be applicable, but not absolutely necessary, since basic limit calculus will show that the limit of f/g as x approaches c must be zero.

Corollary

A simple but very useful consequence of L'Hopital's rule is a well-known criterion for differentiability. It states the following:
suppose that f is continuous at a, and that exists for all x in some open interval containing a, except perhaps for. Suppose, moreover, that exists. Then also exists and
In particular, f' is also continuous at a.

Proof

Consider the functions and . The continuity of f at a tells us that. Moreover, since a polynomial function is always continuous everywhere. Applying L'Hopital's rule shows that.