L'Hôpital's rule

The rule is named after the 17th-century French mathematician 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.

L'Hôpital's rule states that for functions f and g which are defined on an open interval I and differentiable on

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.

Guillaume de l'Hôpital (also written l'Hospital[a]) 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.

[1][b] However, it is believed that the rule was discovered by the Swiss mathematician Johann Bernoulli.

[3] The general form of L'Hôpital's rule covers many cases.

In the second case, the hypothesis that f diverges to infinity is not necessary; in fact, it is sufficient that

appears most commonly in the literature, but some authors sidestep this hypothesis by adding other hypotheses which imply

[c] Another method[5] is to require that both f and g be differentiable everywhere on an interval containing c. All four conditions for L'Hôpital's rule are necessary: 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.

is an open interval is grandfathered in from the hypothesis of the Cauchy's mean value theorem.

Further examples of this type were found by Ralph P. Boas Jr.[7] The requirement that the limit

may exhibit many oscillations of small amplitude but steep slope, which do not affect

But the ratio of the original functions does approach a limit, since the amplitude of the oscillations of

converges to positive or negative infinity, but the justification is then incomplete.

)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: 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 x: Applying L'Hôpital's rule and finding the derivatives with respect to h yields nxn−1 as expected, but this computation requires the use of the very formula that is being proven.

in the first place; a valid proof requires a different method such as the squeeze theorem.

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".

is of the indeterminate form 0 · ∞ 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: The Stolz–Cesàro theorem is a similar result involving limits of sequences, but it uses finite difference operators rather than derivatives.

Consider the parametric curve in the xy-plane with coordinates given by the continuous functions

The last equality follows from the continuity of the derivatives at c. The limit in the conclusion is not indeterminate because

Taylor notes that different proofs may be found in Lettenmeyer (1936) and Wazewski (1949).

ranges over all values between x and c. (The symbols inf and sup denote the infimum and supremum.)

, Cauchy's mean value theorem ensures that for any two distinct points x and y in

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 ±∞.

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.

A simple but very useful consequence of L'Hopital's rule is that the derivative of a function cannot have a removable discontinuity.