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In calculus (an area of mathematics), L'Hôpital's rule uses derivatives to determine otherwise hard to compute limitss. In a more precise statement, it relates to indeterminate forms — if application of the rule leads from an indeterminate form to one that turns out to be determinate, the transformation typically allows one to conclude that the indeterminate form has a limiting value, with the limit equal to the value of the determinate form at the value of the variable in question.
The most common formulation is as follows. If you are trying to determine the limit of some quotient f(x)/g(x), and both the numerator and denominator approach 0 or infinity, then differentiate numerator and denominator and determine the limit of the quotient of the derivatives. If that limit exists, the rule states that its value will be the same as the originally-sought limit.
In symbols,
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2 Proof 3 Of interest |
Here is a case of 0/0:
Examples
Here is a case of ∞/∞:
Sometimes, even limits which don't appear to be quotients can be handled with the same rule:
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 (1692), the first textbook to be written on the differential calculus.
The proof of L'Hôpital's rule depends on Cauchy's mean value theorem.
According to Cauchy's mean value theorem there is a constant in the interval such that:
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 special type of limits are actually easier to compute using the Taylor series expansion.
For example,
"L'Hôpital" is commonly seen spelled both "L'Hospital" and "L'Hôpital" the two spellings being equivalent in the French language.
Proof
Since , we can say that:
If we let , we get:
Therefore
Q.E.D.Of interest