
L'Hôpital's rule (, ), also known as Bernoulli's rule, is a mathematical theorem that allows evaluating
limits of
indeterminate form
Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corres ...
s using
derivatives. Application (or repeated 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
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Guillaume de l'Hôpital. Although the rule is often attributed to de l'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician
Johann Bernoulli
Johann Bernoulli (also known as Jean in French or John in English; – 1 January 1748) was a Swiss people, Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infin ...
.
L'Hôpital's rule states that for functions and which are defined on an open
interval and
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
on
for a (possibly infinite)
accumulation point
In mathematics, a limit point, accumulation point, or cluster point of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood (mathematics), neighbourhood of ...
of , if
and
for all in
, and
exists, then
:
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
Guillaume de l'Hôpital (also written l'Hospital) published this rule in his 1696 book ''
Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes'' (literal translation: ''Analysis of the Infinitely Small for the Understanding of Curved Lines''), the first textbook on
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
. However, it is believed that the rule was discovered by the Swiss mathematician
Johann Bernoulli
Johann Bernoulli (also known as Jean in French or John in English; – 1 January 1748) was a Swiss people, Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infin ...
.
General form
The general form of l'Hôpital's rule covers many cases. Let and be
extended real numbers
In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...
: real numbers, as well as positive and negative infinity. Let be an
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
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). On
, the real-valued functions and are assumed
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
with
. It is also assumed that
, 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, 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
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
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
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
, although it 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 and . 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
In logic and mathematics, contraposition, or ''transposition'', refers to the inference of going from a Conditional sentence, conditional statement into its logically equivalent contrapositive, and an associated proof method known as . The contrap ...
form of the Rule, if
does not exist, then
also does not exist.
Examples
In the following computations, we indicate each application of l'Hôpital's rule 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
In single-variable calculus, the difference quotient is usually the name for the expression
: \frac
which when taken to the Limit of a function, limit as ''h'' approaches 0 gives the derivative of the Function (mathematics), function ''f''. The ...
. Since applying l'Hôpital requires knowing the relevant derivatives, this amounts to
circular reasoning
Circular reasoning (, "circle in proving"; also known as circular logic) is a fallacy, logical fallacy in which the reasoner begins with what they are trying to end with. Circular reasoning is not a formal logical fallacy, but a pragmatic defect ...
or
begging the question
In classical rhetoric and logic, begging the question or assuming the conclusion (Latin: ) is an informal fallacy that occurs when an argument's premises assume the truth of the conclusion. Historically, begging the question refers to a fault i ...
, assuming what is to be proved. For example, consider the proof of the derivative formula for
powers of ''x'':
:
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
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is bounded between two other functions.
The squeeze theorem is used in calculus and mathematical a ...
.
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
logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
s 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:
Stolz–Cesàro theorem
The Stolz–Cesàro theorem is a similar result involving limits of sequences, but it uses finite
difference operator
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s rather than
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s.
Geometric interpretation: parametric curve and velocity vector
Consider the
parametric curve
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters.
In the case ...
in the ''xy''-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
In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...
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.
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s,
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
and
cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
,
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
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''. (The symbols inf and sup denote the
infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
and
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
.)
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''(''x'')-''g''(''y'') is always nonzero for distinct ''x'' and ''y'' in the interval, for if it was not, the
mean value theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
would imply the existence of a ''p'' between ''x'' and ''y'' such that ''g' ''(''p'')=0.
The definition of ''m''(''x'') and ''M''(''x'') 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''(''x'') and ''M''(''x'') 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
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
and
limit inferior are necessary since the existence of the limit of has not yet been established.
It is also the case that
:
and
:
and
In case 1, the
squeeze theorem
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is bounded between two other functions.
The squeeze theorem is used in calculus and mathematical a ...
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''(''x'') diverges to infinity was not used within the proof. This means that if , ''g''(''x''), 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''(''x''): It could even be the case that the limit of ''f''(''x'') does not exist. In this case, L'Hopital's theorem is actually a consequence of Cesàro–Stolz.
In the case when , ''g''(''x''), diverges to infinity as ''x'' approaches ''c'' and ''f''(''x'') 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''(''x'')/''g''(''x'') as ''x'' approaches ''c'' must be zero.
Corollary
A simple but very useful consequence of L'Hopital's rule is that the derivative of a function cannot have a removable discontinuity. That is, 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''.
Thus, if a function is not continuously differentiable near a point, the derivative must have an essential discontinuity at that point.
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
.
See also
*
L'Hôpital controversy
Notes
References
Sources
*
*
*
*
*
{{DEFAULTSORT:Lhopital's Rule
Articles containing proofs
Theorems in calculus
Theorems in real analysis
Limits (mathematics)