l'hôpital's rule wikipedia

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L'Hpital's rule

Guillaume de l'Hpital, after whom this rule is named

Incalculus,l'Hpital's rulepronounced:[lopital](also sometimes spelled l'Hospital's rulewith silent "s" and identical pronunciation), also calledBernoulli's rule, usesderivativesto

help evaluatelimitsinvolvingindeterminate forms. Application (or repeated application) of

the rule often converts an indeterminate form to a determinate form, allowing easy evaluation

of the limit. The rule is named after the 17th-centuryFrenchmathematicianGuillaume de

l'Hpital, who published the rule in his bookAnalyse des Infiniment Petits pour l'Intelligence

des Lignes Courbes(literal translation:Analysis of the Infinitely Small for the Understanding

of Curved Lines) (1696), the first textbook ondifferential calculus.[1][2]

However, it is

believed that the rule was discovered by the Swiss mathematicianJohann Bernoulli.[3]

TheStolz-Cesro theoremis a similar result involving limits of sequences, but it uses finite

difference operatorsrather thanderivatives.

In its simplest form, l'Hpital's rule states that for functionsfand g which aredifferentiable

onI\{c} , whereIis an open interval containing c:

If

, and

exists, and

for allx inIwithx c,

then

.

The differentiation of the numerator and denominator often simplifies the quotient and/orconverts it to a determinate form, allowing the limit to be evaluated more easily.
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General form

The general form of l'Hpital's rule covers many cases. Let c andL beextended real numbers

(i.e., real numbers, positive infinity, or negative infinity). The real valued functionsfand g

are assumed to bedifferentiableon an open interval with endpoint c, and additionally

on the interval. It is also assumed that Thus the rule applies tosituations in which the ratio of the derivatives has a finite or infinite limit, and not to

situations in which that ratio fluctuates permanently asx gets closer and closer to c.

If either

or

then

The limits may also beone-sided limits. In the second case, the hypothesis thatfdivergesto

infinity is not used in the proof (see note at the end of the proof section); 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 appears most commonly in the literature. Some authors

sidestep the hypothesis by adding other hypotheses elsewhere. One method used implicitly in

(Chatterjee 2005, p. 291) is to define the limit of a function with the additional requirement

that the limiting function is defined everywhere on a connected interval with endpoint c[4]

.

Another method appearing in (Krantz 2004, p.79) is to require that bothfand g are

differentiable everywhere on an interval containing c.

Requirement that limit exist

The requirement that the limit
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exist is essential. Without this condition, it may be the case that and/or exhibits

undampened oscillations asx approaches c. If this happens, then l'Hpital's rule does not

apply. For example, if and , then

this expression does not approach a limit, since the cosine function oscillates between 1 and

1. But working with the original functions, can be shown to exist:

.

Examples

Here is an example involving thesinc functionand the indeterminate form 00:

Alternatively, just observe that the limit is the definition of the derivative of thesine

function at zero.

This is a more elaborate example involving 00. Applying l'Hpital's rule a single timestill results in an indeterminate form. In this case, the limit may be evaluated by

applying the rule three times:
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This example involves 00. Suppose that b > 0. Then

Here is another example involving 00:

This example involves . Assume n is a positive integer. Then

Repeatedly apply l'Hpital's rule until the exponent is zero to conclude that the limit is

zero.

Here is another example involving :

Here is an example involving the impulse response of araised-cosine filterand 00:

One can also use l'Hpital's rule to prove the following theorem. Iff is continuous atx, then

Sometimes l'Hpital's rule is invoked in a tricky way: supposef(x) +f(x) convergesasx . It follows:
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and so, if exists and is finite, then

Note that the above proof assumes that exf(x) converges to positive or negative infinity

asx , allowing us to apply L'Hpital's rule. The results however still hold even

though the proof is not entirely complete.

L'Hpital's rule can be used to find the limiting form of a function. In the field ofchoice under uncertainty, thevon Neumann-Morgenstern utilityfunction

with , defined overx>0, is said to have constantrelative risk aversionequal to. But unit relative risk aversion cannot be expressed directly with this expression,

since as approaches 1 the numerator and denominator both approach zero. However,

a single application of l'Hpital's rule allows this case to be expressed as

Complications

Sometimes l'Hpital's rule does not lead to an answer in a finite number of steps unless a

transformation of variables is applied. Examples include the following:

Two applications can lead to a return to the original expression that was to beevaluated:

This situation can be dealt with by substituting and noting thaty goes to

infinity asx goes to infinity; with this substitution, this problem can be solved with a

single application of the rule:

An arbitrarily large number of applications may never lead to an answer even withoutrepeating:

This situation too can be dealt with by a transformation of variables, in this case

:

Other indeterminate forms
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Other indeterminate forms, such as 1, 00,

0, 0 , and , can sometimes be evaluated

using l'Hpital's rule. For example, to evaluate a limit involving , convert the difference

of two functions to a quotient:

where l'Hpital's rule was applied in going from (1) to (2) and then again in going from (3) to

(4).

l'Hpital's rule can be used on indeterminate forms involvingexponentsby usinglogarithms

to "move the exponent down". Here is an example involving the indeterminate form 00:

It is valid to move the limit inside theexponential functionbecause the exponential function

iscontinuous. Now the exponent has been "moved down". The limit limx0+

(x lnx) is of

the indeterminate form 0 (-), but as shown in an example above, l'Hpital's rule may be

used to determine that

Thus

Other methods of evaluating limits

Although l'Hpital's rule is a powerful way of evaluating otherwise hard-to-evaluate limits, it

is not always the easiest way. Consider
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This limit may be evaluated using l'Hpital's rule:

It is valid to move the limit inside thecosinefunction because the cosine function iscontinuous.

But a simpler way to evaluate this limit is to use a substitution.y = 1/x. As |x| approaches

infinity,y approaches zero. So,

The final limit may be evaluated using l'Hpital's rule or by noting that it is the definition of

the derivative of the sine function at zero.

Still another way to evaluate this limit is to use aTaylor seriesexpansion:

For |x| 1, the expression in parentheses is bounded, so the limit in the last line is zero.

Geometric interpretation

Consider the curve in the plane whosex-coordinate is given by g(t) and whosey-coordinate is

given byf(t), i.e.
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Supposef(c) = g(c) = 0. The limit of the ratiof(t)/g(t) as t c is the slope of tangent to the

curve at the point [0, 0]. The tangent to the curve at the point tis given by [g(t),f(t)].

L'Hpital's rule then states that the slope of the tangent at 0 is the limit of the slopes of

tangents at the points approaching zero.

Proof of l'Hpital's rule

Special case

The proof of l'Hpital's rule is simple in the case wherefand g arecontinuously

differentiableat the point c and where a finite limit is found after the first round of

differentiation. It is not a proof of the general l'Hpital'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 (e.g.polynomials,sineandcosine,exponential

functions), it is a special case worthy of attention.

Suppose thatfand g are continuously differentiable at a real number c, 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 c. The limit in the conclusion is not

indeterminate because .

The proof of a more general version of L'Hpital's rule is given below.

General proof

The following proof is due to (Taylor 1952), where a unified proof for the 0/0 and /

indeterminate forms is given. Taylor notes that different proofs may be found in

(Lettenmeyer 1936) and (Wazewski 1949).

Letfand g be functions satisfying the hypotheses in theGeneral formsection. Let be theopen 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[5]

.

For eachx in the interval, define and asranges over all values betweenx and c. (The symbols inf and sup denote theinfimumand

supremum.)

From the differentiability offand g on ,Cauchy's mean value theoremensures that for any

two distinct pointsx andy in there exists a betweenx andy such that
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. Consequently for all

choices of distinctx andy in the interval. The value g(x)-g(y) is always nonzero for distinctx

andy in the interval, for if it was not, themean value theoremwould imply the existence of a

p betweenx andy such that g'(p)=0.

The definition ofm(x) andM(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) andM(x) will establish

bounds on the ratiof/g.

Case 1:

For anyx in the interval , and pointy betweenx and c,

and therefore asy approaches c, and become zero, and so

.

Case 2:

For anyx in the interval , define . For any pointy

betweenx and c, we have

.

Asy approaches c, both and become zero, and therefore

Thelimit superiorandlimit inferiorare necessary since the existence of the limit off/g has

not yet been established.

We need the facts that
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and

and

.

In case 1, theSqueeze theorem, establishes that exists and is equal toL. In the

case 2, and the Squeeze theorem again asserts that ,

and so the limit exists and is equal toL. This is the result that was to be proven.

Note: In case 2 we did not use the assumption thatf(x) diverges to infinity within the proof.

This means that if |g(x)| diverges to infinity asx approaches c and bothfand g satisfy the

hypotheses of l'Hpital's rule, then no additional assumption is needed about the limit off(x):

It could even be the case that the limit off(x) does not exist.

In the case when |g(x)| diverges to infinity asx approaches c andf(x) converges to a finitelimit at c, then l'Hpital's rule would be applicable, but not absolutely necessary, since basic

limit calculus will show that the limit off(x)/g(x) asx approaches c must be zero.

Indeterminate form

Incalculusand other branches ofmathematical analysis, an indeterminate form is an

algebraic expressionobtained in the context oflimits. Limits involving algebraic operations

are often performed by replacing subexpressions by their limits; if the expression obtained

after this substitution does not give enough information to determine the original limit, it is

known as an indeterminate form. The indeterminate forms include 00, 0/0, 1, , /,

0 , and 0.

Discussion

The most common example of an indeterminate form is 0/0. Asx approaches 0, the ratios

x/x3,x/x, andx

2/x go to , 1, and 0 respectively. In each case, however, if the limits of the

numerator and denominator are evaluated and plugged into the division operation, the

resulting expression is 0/0. So (roughly speaking) 0/0 can be 0, or , or it can be 1 and, in

fact, it is possible to construct similar examples converging to any particular value. That is

why the expression 0/0 is indeterminate.
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More formally, the fact that thefunctionsfand g both approach 0 asx approaches somelimit

pointc is not enough information to evaluate thelimit

That limit could converge to any number, or diverge to infinity, or might not exist, depending

on what the functionsfand g are.

In some theories a value may be defined even where the function is discontinuous. For

example |x|/x is undefined forx = 0 inreal analysis. However it is thesign functionwith

sgn(0) = 0 when consideringFourier seriesorhyperfunctions.

Not every undefined algebraic expression is an indeterminate form. For example, the

expression 1/0 is undefined as areal numberbut is not indeterminate. This is because any

limit that gives rise to this form willdiverge to infinity.

An expression representing an indeterminate form may sometimes be given a numerical value

in settings other than the computation of limits. The expression 00

is defined as 1 when it

represents anempty product. In the theory ofpower series, it is also often treated as 1 by

convention, to make certainformulasmore concise. (See the section "Zero to the zero power"

in the article onexponentiation.) In the context ofmeasure theory, it is usual to take to be

0.

Some examples and non-examples

The form 0/0

(1)

(2)

(3)
http://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Limit_pointhttp://en.wikipedia.org/wiki/Limit_pointhttp://en.wikipedia.org/wiki/Limit_pointhttp://en.wikipedia.org/wiki/Limit_pointhttp://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/Real_analysishttp://en.wikipedia.org/wiki/Real_analysishttp://en.wikipedia.org/wiki/Real_analysishttp://en.wikipedia.org/wiki/Sign_functionhttp://en.wikipedia.org/wiki/Sign_functionhttp://en.wikipedia.org/wiki/Sign_functionhttp://en.wikipedia.org/wiki/Fourier_serieshttp://en.wikipedia.org/wiki/Fourier_serieshttp://en.wikipedia.org/wiki/Fourier_serieshttp://en.wikipedia.org/wiki/Hyperfunctionhttp://en.wikipedia.org/wiki/Hyperfunctionhttp://en.wikipedia.org/wiki/Hyperfunctionhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Divergence_to_infinityhttp://en.wikipedia.org/wiki/Divergence_to_infinityhttp://en.wikipedia.org/wiki/Divergence_to_infinityhttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Series_%28mathematics%29http://en.wikipedia.org/wiki/Series_%28mathematics%29http://en.wikipedia.org/wiki/Series_%28mathematics%29http://en.wikipedia.org/wiki/Formulahttp://en.wikipedia.org/wiki/Formulahttp://en.wikipedia.org/wiki/Formulahttp://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_powerhttp://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_powerhttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Measure_theoryhttp://en.wikipedia.org/wiki/Measure_theoryhttp://en.wikipedia.org/wiki/Measure_theoryhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_sin_x_over_x_close.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x2_over_x.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x_over_x.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_sin_x_over_x_close.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x2_over_x.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x_over_x.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_sin_x_over_x_close.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x2_over_x.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x_over_x.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_sin_x_over_x_close.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x2_over_x.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x_over_x.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_sin_x_over_x_close.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x2_over_x.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_x_over_x.gifhttp://en.wikipedia.org/wiki/Measure_theoryhttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_powerhttp://en.wikipedia.org/wiki/Formulahttp://en.wikipedia.org/wiki/Series_%28mathematics%29http://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Divergence_to_infinityhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Hyperfunctionhttp://en.wikipedia.org/wiki/Fourier_serieshttp://en.wikipedia.org/wiki/Sign_functionhttp://en.wikipedia.org/wiki/Real_analysishttp://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/Limit_pointhttp://en.wikipedia.org/wiki/Limit_pointhttp://en.wikipedia.org/wiki/Function_%28mathematics%29


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(4)

The indeterminate form 0/0 is particularly common incalculusbecause it often arises in the

evaluation ofderivativesusing their limit definition.

As mentioned above,

while

This is enough to show that 0/0 is an indeterminate form. Other examples with this

indeterminate form include

and

Direct substitution of the number thatx approaches into any of these expressions leads to theindeterminate form 0/0, but the limits take many different values. In fact, any desired valueA

can be obtained for this indeterminate form as follows:

Furthermore, the value infinity can also be obtained (in the sense of divergence to infinity):

The form 00

The indeterminate form 00

has been discussed since at least 1834.[1]

The following examples

illustrate that the form is indeterminate:
http://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Indeterminate_form#cite_note-0http://en.wikipedia.org/wiki/Indeterminate_form#cite_note-0http://en.wikipedia.org/wiki/Indeterminate_form#cite_note-0http://en.wikipedia.org/wiki/File:Indeterminate_form_-_complicated.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_complicated.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_complicated.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_complicated.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_complicated.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_complicated.gifhttp://en.wikipedia.org/wiki/File:Indeterminate_form_-_complicated.gifhttp://en.wikipedia.org/wiki/Indeterminate_form#cite_note-0http://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Calculus


13/15

Thus, in general, knowing that and is not sufficient to

calculate the limit

If the functionsfand g areanalyticandfis not identically zero in aneighbourhoodofc on thecomplex plane, then the limit off(z)

g(z) will always be 1. This also holds for real functions,

butfmust not be negative in the domain of the limit; alternatively,fcan be the absolute value

of an analytic function.[2]

If the pair (f(x), g(x)) remains between two linesy = mx andy =Mx, where m andMare

positive numbers, asx approaches c andf(x) and g(x) approaches 0, then the limit is always 1.

In many settings other than when evaluating limits 00

is taken to be defined as 1 even though

it is an indeterminate form; see the sectionzero to the zero powerin the article on

exponentiation. One justification for this is provided by the preceding result. Another is that

in power series, such as

whenx = 0, then the term in which n = 0 has the correct value only if 00

= 1. Yet another is

that incombinatorialproblems one must sometimes take 00

to be anempty product.

Undefined forms that are not indeterminate

The expression 1/0 is not commonly regarded as an indeterminate form because there is not

an infinite range of values thatf/g could approach. Specifically, iffapproaches 1 and g

approaches 0, thenfand g may be chosen so that: (1)f/gapproaches +, (2)f/g approaches -

, or (3) the limit fails to exist. In each case the absolute value |f/g| approaches +, and so thequotientf/g must diverge, in the sense of theextended real numbers. (In the framework of the

real projective line, the limit is theunsigned infinity in all three cases.)

Similarly, any expression of the form a/0, with a0 (including a=+ and a=-), is not an

indeterminate since a quotient giving rise to such an expression will always diverge.

Also, 0

is sometimes incorrectly thought to be indeterminate, but is not indeterminate:

0+

=0, and 0-

is equivalent to 1/0.

Evaluating indeterminate forms
http://en.wikipedia.org/wiki/Analytic_functionhttp://en.wikipedia.org/wiki/Analytic_functionhttp://en.wikipedia.org/wiki/Analytic_functionhttp://en.wikipedia.org/wiki/Neighbourhood_%28topology%29http://en.wikipedia.org/wiki/Neighbourhood_%28topology%29http://en.wikipedia.org/wiki/Neighbourhood_%28topology%29http://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Indeterminate_form#cite_note-1http://en.wikipedia.org/wiki/Indeterminate_form#cite_note-1http://en.wikipedia.org/wiki/Indeterminate_form#cite_note-1http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_powerhttp://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_powerhttp://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_powerhttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Combinatoricshttp://en.wikipedia.org/wiki/Combinatoricshttp://en.wikipedia.org/wiki/Combinatoricshttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Extended_real_numberhttp://en.wikipedia.org/wiki/Extended_real_numberhttp://en.wikipedia.org/wiki/Extended_real_numberhttp://en.wikipedia.org/wiki/Real_projective_linehttp://en.wikipedia.org/wiki/Real_projective_linehttp://en.wikipedia.org/wiki/Point_at_infinityhttp://en.wikipedia.org/wiki/Point_at_infinityhttp://en.wikipedia.org/wiki/Point_at_infinityhttp://en.wikipedia.org/wiki/Point_at_infinityhttp://en.wikipedia.org/wiki/Real_projective_linehttp://en.wikipedia.org/wiki/Extended_real_numberhttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Combinatoricshttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_powerhttp://en.wikipedia.org/wiki/Indeterminate_form#cite_note-1http://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Neighbourhood_%28topology%29http://en.wikipedia.org/wiki/Analytic_function


14/15

The indeterminate nature of a limit's form does notimply that the limit does not exist, as

many of the examples above show. In many cases, algebraic elimination,L'Hpital's rule, or

other methods can be used to manipulate the expression so that the limit can be evaluated.

For example, the expressionx2/x can be simplified tox at any point other thanx = 0. Thus,

the limit of this expression asx approaches 0 (which depends only on points near0, not atx =0 itself) is the limit ofx, which is 0. Most of the other examples above can also be evaluated

using algebraic simplification.

L'Hpital's rule is a general method for evaluating the indeterminate forms 0/0 and /. This

rule states that (under appropriate conditions)

wheref'and g'are thederivativesoffand g. (Note that this rule does notapply to forms like/0, 1/0, and so on; but these forms are not indeterminate either.) With luck, these derivatives

will allow one to perform algebraic simplification and eventually evaluate the limit.

L'Hpital's rule can also be applied to other indeterminate forms, using first an appropriate

algebraic transformation. For example, to evaluate the form 00:

The right-hand side is of the form /, so L'Hpital's rule applies to it. Notice that thisequation is valid (as long as the right-hand side is defined) because thenatural logarithm(ln)

is acontinuous function; it's irrelevant how well-behavedfand g may (or may not) be as long

asfis asymptotically positive.

Although L'Hpital's rule applies both to 0/0 and to /, one of these may be better than the

other in a particular case (because of the possibilities for algebraic simplification afterwards).

You can change between these forms, if necessary, by transformingf/g to (1/g)/(1/f).

List of indeterminate forms

The following table lists the indeterminate forms for the standardarithmetic operationsand

the transformations for applying l'Hpital's rule.

Indeter

minate

form

Conditions Transformation to 0/0 Transformation to /

0/0

/
http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rulehttp://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rulehttp://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rulehttp://en.wikipedia.org/wiki/Derivative_%28calculus%29http://en.wikipedia.org/wiki/Derivative_%28calculus%29http://en.wikipedia.org/wiki/Derivative_%28calculus%29http://en.wikipedia.org/wiki/Natural_logarithmhttp://en.wikipedia.org/wiki/Natural_logarithmhttp://en.wikipedia.org/wiki/Natural_logarithmhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Arithmetic_operationhttp://en.wikipedia.org/wiki/Arithmetic_operationhttp://en.wikipedia.org/wiki/Arithmetic_operationhttp://en.wikipedia.org/wiki/Arithmetic_operationhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Natural_logarithmhttp://en.wikipedia.org/wiki/Derivative_%28calculus%29http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule


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0

1

00

0

l'hôpital's rule wikipedia

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