In calculus, l'Hôpital's rule uses derivatives to help compute limits with indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy computation of the limit. The rule is named after the 17th century French mathematician Guillaume François Antoine, Marquis de l'Hôpital (1661 - 1704), who published the rule in his book Analyse des infiniment petits pour l'intelligence des lignes courbes (1696), the first book to be written on the differential calculus. It is likely, however, that the result was originally due to Johann Bernoulli, upon whose lectures the text was largely based. For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ... In mathematics, a number of the expressions that may be encountered in calculus and occasionally elsewhere are considered to be indeterminate forms, and must be treated as symbolic only, until more careful discussion has taken place. ... (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... Guillaume François Antoine, Marquis de lHôpital (1661 - February 2, 1704) was a French mathematician. ... The year 1696 had the earliest equinoxes and solstices for 400 years in the Gregorian calendar, because this year is a leap year and the Gregorian calendar would have behaved like the Julian calendar since March 1500 had it have been in use that long. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Johann Bernoulli Johann Bernoulli (Basel, July 27, 1667 - January 1, 1748) was a Swiss mathematician. ...

Overview

When determining the limit of a quotient f(x)/g(x) when both the numerator and denominator approach 0 or the denominator approaches infinity, l'Hôpital's rule states that differentiation of both the numerator and denominator does not change the limit. This differentiation, however, often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be determined more easily.

Symbolically, if and

then

Please note the requirement that the limit f′/g′ exists. Differentiation of limits form can sometimes lead to limits that don't exist. In that case, l'Hôpital's rule can't be applied. In real calculations, one would just use the rule "on a good word" and then if he manages to solve the resulting limit, he can conclude that it was possible to use l'Hôpital's theorem.

Examples

• Here is an example involving 0/0 in the sinc function:

However, it is simpler to observe that this limit is just the definition of the derivative of sin(x) at x = 0.

In fact this particular limit is needed in the most usual proof that the derivative of sin(x) is cos(x), but we cannot use l'Hôpital's rule to do this, as it would produce a circular argument.

• Here is a more elaborate example involving the indeterminate form 0/0. Applying the rule a single time still results in an indeterminate form. In this case, the limit may be evaluated by applying l'Hôpital's rule three times:

• Here is another case involving 0/0.

• Here is a case of ∞/∞:

• This one involves ∞/∞. Assume n is a positive integer.

Iterate the above until the exponent is 0. Then one sees that the limit is 0.

• This one involves ∞/∞.

• This is known as the raised cosine function.

The sinc function sinc(x) from x=-8π to 8π. ... Begging the question, in modern popular usage, is often used synonymously for raising the question. However the original meaning is quite different: it described a type of logical fallacy (also called petitio principii) in which the evidence given for a proposition as much needs to be proved as the proposition... In mathematics, a number of the expressions that may be encountered in calculus and occasionally elsewhere are considered to be indeterminate forms, and must be treated as symbolic only, until more careful discussion has taken place. ... Raised cosine is a function commonly used in wireless communications. ...

Proof

The most common proof of l'Hôpital's rule uses Cauchy's mean value theorem. For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ...

1) The case when

First, we expand continuously (or redefine) f(x) and g(x) by 0 for x = c. This doesn't change the limit since the limit doesn't depend on the value in the point (by definition).

According to Cauchy's mean value theorem there is a constant ξ in c < ξ < c + h such that:

Since f(c) = g(c) = 0,

If then and

2) The case when

Let x < y < x + h. Then using Cauchy's mean value theorem: For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ...

We rewrite that in the form

and then by the discussion of all the three cases

we show that the limit of f(x)/g(x) tends to the same when and .

Other proofs

There are more intuitive proofs of the rule. If

tends to the indeterminate form 0/0, then the rule can be proven with a local linearity argument. If it tends to the indeterminate form , then this can be converted to 0/0 form using the identity : By assuming this limit equals L, and taking the derivative of the numerator and denominator, it can be proven that .

Other applications

Many other indeterminate forms, such as , , and can be calculated using l'Hôpital's rule.

For example, to handle a case of , the difference of two functions is converted to a quotient:

Other methods of computing limits

Although l'Hôpital's rule's rule is a powerful way of computing otherwise hard-to-compute limits, it is not always the easiest. Some limits are actually easier to compute using the Taylor series expansion. As the degree of the taylor series rises, it approaches the correct function. ...

For example,

Logical circularity

In some cases it may constitute circular reasoning to use l'Hôpital's rule to evaluate such limits as Begging the question, in modern popular usage, is often used synonymously for raising the question. However the original meaning is quite different: it described a type of logical fallacy (also called petitio principii) in which the evidence given for a proposition as much needs to be proved as the proposition...

If one uses the evaluation of the limit above for the purpose of proving that

and one uses l'Hôpital's rule and the fact that

in the evaluation of the limit, then one's reasoning is circular and therefore fallacious.

See also

• Infinity tricks