L'Hôpital's rule ( ), is a mathematical theorem used for evaluating the limit of a quotient of two functions, each of which tends to zero or infinity, by taking each function's derivative. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital, who published it in his 1696 textbook after learning it from his tutor, the Swiss mathematician Johann Bernoulli.
For two functions and , under most circumstances the limit of their quotient can be evaluated as the quotient of the limits: . However, if both limits tend to zero or if both tend to infinity, this method cannot be applied because the and are indeterminate forms, meaning they are not well defined. L'Hôpital's rule states that in such cases, under the assumption that there exists an open interval containing in which never equals zero except possibly at ,whenever the limit on the right-hand side exists. Here, and are the derivatives of and .
The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be directly evaluated by continuity.
History Johann Bernoulli was the original discoverer of this result for the indeterminate form . He sent it to Guillaume de l'Hôpital (also written l'Hospital) in a letter dated July 22, 1694. Previously, during his time in Paris, Bernoulli signed a contract with l'Hôpital, in which he agreed to teach the new calculus to l'Hôpital, to inform him of his own mathematical discoveries, which l'Hôpital might use as he pleased, and to refrain from sharing his notes with others. In return, l'Hôpital paid Bernoulli an annual allowance, which dramatically improved Bernoulli's financial situation. L'Hôpital published this rule in his 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes (). Its original presentation was geometric and L'Hôpital showcased many of the same examples used by Bernoulli. As the first textbook on the differential calculus to be printed, it dominated much of the eighteenth century and helped popularize new mathematics across Europe. The original result was subsequently extended; these are now collectively referred to as l'Hôpital's rules.
General form The general form of l'Hôpital's rule covers many cases. Let and be extended real numbers: real numbers, as well as positive and negative infinity. Let be an open interval containing (for a two-sided limit) or an open interval with endpoint (for a one-sided limit, or a limit at infinity if is infinite).
Assumption 1: On , the real-valued functions and are differentiable with .
Assumption 2: , a finite or infinite limit.
If either
or
then
Although we have written throughout, the limits may also be one-sided limits ( or ), when is a finite endpoint of .
In the second case b), the hypothesis that diverges to infinity is not necessary; in fact, it is sufficient that
The hypothesis that appears most commonly in the literature, but some authors sidestep this hypothesis by adding other hypotheses which imply . For example, one may require in the definition of the limit that the function must be defined everywhere on an interval . Another method is to require that both and be differentiable everywhere on an interval containing .
Necessity of conditions: Counterexamples
All four conditions for l'Hôpital's rule are necessary:
Indeterminacy of form: or ; Differentiability of functions: and are differentiable on an open interval except possibly at the limit point in ; Non-zero derivative of denominator: for all in with ; Existence of limit of the quotient of the derivatives: exists.
Where one of the above conditions is not satisfied, l'Hôpital's rule is not valid in general, and its conclusion may be false in certain cases.
1. Form is not indeterminate The necessity of the first condition can be seen by considering the counterexample where the functions are and and the limit is .
The first condition is not satisfied for this counterexample because and . This means that the form is not indeterminate.
The second and third conditions are satisfied by and . The fourth condition is also satisfied with
But the conclusion fails, since
2. Differentiability of functions Differentiability of functions is a requirement because if a function is not differentiable, then the derivative of the function is not guaranteed to exist at each point in . The fact that is an open interval is grandfathered in from the hypothesis of the Cauchy's mean value theorem. The notable exception of the possibility of the functions being not differentiable at exists because l'Hôpital's rule only requires the derivative to exist as the function approaches ; the derivative does not need to be taken at .
For example, let , , and . In this case, is not differentiable at . However, since is differentiable everywhere except , then still exists. Thus, since and exists, l'Hôpital's rule still holds.
3. Derivative of denominator is zero The necessity of the condition that near (except possibly at ) 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 , while the original expression before cancellation of is undefined at infinitely many points. Further examples of this type were found by Ralph P. Boas Jr.
4. Limit of derivatives does not exist The requirement that the limit exists is essential; if it does not exist, the original limit may nevertheless exist. Indeed, as approaches , the functions or may exhibit many oscillations of small amplitude but steep slope, which do not affect but do prevent the convergence of .
For example, if , and , then
which does not approach a limit since cosine oscillates infinitely between 1 and −1. But the ratio of the original functions does approach a limit, since the amplitude of the oscillations of becomes small relative to :
In a case such as this, all that can be concluded is that
so that if the limit of exists, then it must lie between the inferior and superior limits of . In the example, 1 does indeed lie between 0 and 2.)
Note also that by the contrapositive form of the Rule, if does not exist, then also does not exist.
Examples In the following computations, each application of l'Hôpital's rule is indicated by the symbol .
Here is a basic example involving the exponential function, which involves the indeterminate form at : This is a more elaborate example involving . Applying l'Hôpital's rule a single time still results in an indeterminate form. In this case, the limit may be evaluated by applying the rule three times: Here is an example involving : Repeatedly apply l'Hôpital's rule until the exponent is zero (if is an integer) or negative (if is fractional) to conclude that the limit is zero. Here is an example involving the indeterminate form (see below), which is rewritten as the form : Here is an example involving the mortgage repayment formula and . Let be the principal (loan amount), the interest rate per period and the number of periods. When is zero, the repayment amount per period is (since only principal is being repaid); this is consistent with the formula for non-zero interest rates: One can also use l'Hôpital's rule to prove the following theorem. If is twice-differentiable in a neighborhood of and its second derivative is continuous on this neighborhood, then Sometimes l'Hôpital's rule is invoked in a tricky way: suppose converges as and that converges to positive or negative infinity. Then:and so, exists and (This result remains true without the added hypothesis that converges to positive or negative infinity, but the justification is then incomplete.)
Complications Sometimes L'Hôpital's rule does not reduce to an obvious limit in a finite number of steps, unless some intermediate simplifications are applied. Examples include the following:
Two applications can lead to a return to the original expression that was to be evaluated:
This situation can be dealt with by substituting and noting that goes to infinity as goes to infinity; with this substitution, this problem can be solved with a single application of the rule:
Alternatively, the numerator and denominator can both be multiplied by at which point L'Hôpital's rule can immediately be applied successfully
An arbitrarily large number of applications may never lead to an answer even without repeating:
This situation too can be dealt with by a transformation of variables, in this case :
Again, an alternative approach is to multiply numerator and denominator by before applying L'Hôpital's rule:
A common logical fallacy is to use L'Hôpital's rule to prove the value of a derivative by computing the limit of a difference quotient. Since applying l'Hôpital requires knowing the relevant derivatives, this amounts to circular reasoning or begging the question, assuming what is to be proved. For example, consider the proof of the derivative formula for powers of :
Applying L'Hôpital's rule and finding the derivatives with respect to yields as expected, but this computation requires the use of the very formula that is being proven. Similarly, to prove , applying L'Hôpital requires knowing the derivative of at , which amounts to calculating in the first place; a valid proof requires a different method such as the squeeze theorem.
Other indeterminate forms Other indeterminate forms, such as , , , , and , can sometimes be evaluated using L'Hôpital's rule. We again indicate applications of L'Hopital's rule by .
For example, to evaluate a limit involving , convert the difference of two functions to a quotient:
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 this function is continuous. Now the exponent has been "moved down". The limit is of the indeterminate form dealt with in an example above: L'Hôpital may be used to determine that
Thus
The following table lists the most common indeterminate forms and the transformations which precede applying l'Hôpital's rule:
Indeterminate form with and Conditions Transformation to
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: parametric curve and velocity vector Consider the parametric curve in the -plane with coordinates given by the continuous functions and , the locus of points , and suppose . The slope of the tangent to the curve at is the limit of the ratio as . The tangent to the curve at the point is the velocity vector with slope . L'Hôpital's rule then states that the slope of the curve at the origin () is the limit of the tangent slope at points approaching 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. This is only a special case of L'Hôpital's rule, because it only applies to functions satisfying stronger conditions than required by the general rule. However, many common functions have continuous derivatives (e.g. polynomials, sine and cosine, exponential functions), so this special case covers most applications.
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