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In calculus, l'Hôpital's rule (pronounced: [lopiˈtal]) uses derivatives to help evaluate limits involving indeterminate 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-century French mathematician Guillaume de l'Hôpital (also written l'Hospital), who published the 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. However, it is believed that the rule was discovered by the Swiss mathematician Johann Bernoulli.
In its simplest form, l'Hôpital's rule states that for functions f and g which are differentiable on an open interval I except possibly at a point c contained in I:
The differentiation of the numerator and denominator often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be evaluated more easily.
The general form of l'Hôpital's rule covers many cases. Let c and L be extended real numbers (i.e., real numbers, positive infinity, or negative infinity). The real valued functions f and g are assumed to be differentiable on an open interval with endpoint c, and additionally on the interval. It is also assumed that Thus the rule applies to situations in which the ratio of the derivatives has a finite or infinite limit, and not to situations in which that ratio fluctuates permanently as x gets closer and closer to c.
The limits may also be one-sided limits. In the second case, the hypothesis that f diverges to 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. Another method appearing in (Krantz 2004, p.79) is to require that both f and g are differentiable everywhere on an interval containing c.
The requirement that the limit
exist is essential. Without this condition, it may be the case that and/or exhibits undampened oscillations as x approaches c. If this happens, then l'Hôpital'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:
Sometimes l'Hôpital'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:
A common pitfall is using l'Hôpital's rule with some circular reasoning to compute a derivative via a difference quotient. For example, consider the task of proving the derivative formula for powers of x:
Applying l'Hôpital's rule and finding the derivatives with respect to h of the numerator and the denominator yields n xn - 1 as expected. However, differentiating the numerator required the use of the very fact that is being proven. This is an example of begging the question, since one may not assume the fact to be proven during the course of the proof.
Other indeterminate forms, such as 1∞, 00, ∞0, 0 × ∞, and ∞ − ∞, can sometimes be evaluated using l'Hôpital's rule. For example, to evaluate a limit involving ∞ − ∞, convert the difference of two functions to a quotient:
where l'Hôpital's rule was applied in going from (1) to (2) and then again in going from (3) to (4).
It is valid to move the limit inside the exponential function because the exponential function is continuous. Now the exponent has been "moved down". The limit limx→0+ (x ln x) is of the indeterminate form 0 × (−∞), but as shown in an example above, l'Hôpital's rule may be used to determine that
Although l'Hôpital's rule is a powerful way of evaluating otherwise hard-to-evaluate limits, it is not always the easiest way. Consider
This limit may be evaluated using l'Hôpital's rule:
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'Hôpital'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 a Taylor series expansion:
For |x| ≥ 1, the expression in parentheses is bounded, so the limit in the last line is zero.
Consider the curve in the plane whose x-coordinate is given by g(t) and whose y-coordinate is given by f(t), with both functions continuous, i.e., the locus of points of the form
Suppose f(c) = g(c) = 0. The limit of the ratio f(t)/g(t) as t → c is the slope of the tangent to the curve at the point [f(c), g(c)] = [0, 0]. The tangent to the curve at the point [g(t), f(t)] is given by [g′(t), f ′(t)]. L'Hôpital's rule then states that the slope of the tangent when t = c is the limit of the slope of the tangent to the curve as the curve approaches the origin, provided that this is defined.
The proof of l'Hôpital's rule is simple in the case where f and g are continuously differentiable at 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'Hôpital'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, sine and cosine, exponential functions), it is a special case worthy of attention.
Suppose that f and 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'Hôpital's rule is given below.
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).
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 .
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 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 f/g.
For any x in the interval , and point y between x and c,
and therefore as y approaches c, and become zero, and so
For any x in the interval , define . For any point y between x and c, we have
As y approaches c, both and become zero, and therefore
We need the facts that
In case 1, the Squeeze theorem, 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.
Note: In case 2 we did not use the assumption that f(x) diverges to infinity 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 (see proof at http://www.imomath.com/index.php?options=686).
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.
A simple but very useful consequence of l'Hopital's rule is a well-known criterion for differentiability. It states the following: suppose that f is continuous at a, and that exists for all x in some interval containing a, except perhaps for . Suppose, moreover, that exists. Then also exists, and
It suffices to consider the functions and . The continuity of f at a tells us that ; obviously also , since a polynomial function is always continuous everywhere. Applying l'Hopital's rule we conclude that .