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L'Hôpital's rule
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In calculus, l'Hôpital's rule (also spelled l'Hospital; also called Bernoulli'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 de l'Hôpital, who published the rule in his book l'Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes (literal translation: Analysis of the Infinitely Small to Understand Curved Lines) (1696), the first book about differential calculus.
The Stolz-Cesàro theorem is a similar result involving limits of sequences, but it uses finite difference operators rather than derivatives.
Overview
Formal statement
When determining the limit of a quotient when both f and g approach 0, or f and g approach infinity, l'Hôpital's rule states that if converges, then converges, and to the same limit. This differentiation often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be determined more easily.
Symbolically let . Suppose that , that
and that for all in an open interval (a,b) containing c (or with if or with if ). If
then
l'Hôpital's rule also holds for one-sided limits.
Basic indeterminate forms (all others reduce to these):
Other indeterminate forms:
Note the requirement that the limit exists. Differentiation of limits of this form can sometimes lead to limits that do not exist. In that case, l'Hôpital's rule cannot be applied. For instance if and , then
does not exist, whereas
In practice one often uses the rule and, if the resulting limit exists, concludes that it was legitimate to use l'Hôpital's rule.
Note also the requirement that the derivative of g not vanish throughout an entire interval containing the point c. Without such a hypothesis, the conclusion is false. Thus one must not use l'Hôpital's rule if the denominator oscillates wildly near the point where one is trying to find the limit. For example if and , then
whereas
does not exist since fluctuates between e-1 and e.
Examples
- Here is an example involving the sinc function , which has the form 0/0:
- 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 a classic case involving 0/0. Suppose , then
- Here is another case involving 0/0:
- This one involves 8/8. Assume n is a positive integer.
- Iterate the above until the exponent is 0. Then one sees that the limit is 0.
- This one also involves 8/8:
- The previous result can be used the following case of the indeterminate form : To compute , we rewrite as and get
Proofs of l'Hôpital's rule
Proof by Cauchy's mean value theoremThe most common proof of l'Hôpital's rule uses Cauchy's mean value theorem.
With the indeterminate form 0 over 0This is the case where and .
First, define (or redefine) and . This does not change the limit, since the limit does not depend on the value at the point (by definition).
Take to be some point close to .
According to Cauchy's mean value theorem there is a point between and such that:
Since ,
If , then also and
as required.
With the indeterminate form infinity over infinityThis is the case where .
In this case, we only sketch the argument.
Take points satisfying or .
We then use the Cauchy's mean value theorem:
where is some point between and .
We rewrite that in the form
We can then first take so close to that the ratio
is very close to its limit as for all points between and . Next, as we let , the terms
and go to , because . The above equality therefore shows that for close to , we have close to . Therefore,
as needed.
Other applications
Many other indeterminate forms, such as 18, 00, 80, and can be calculated using l'Hôpital's rule.
For example, to handle a case of , the difference of two functions are converted to a quotient:
The rule can be used on indeterminate forms involving exponents by using logarithms to "move the exponent down." For example, with the indeterminate form 00:
;;;
;;;
It is valid to move the limit inside the exponential function, because it is a continuous function. Now the exponent x has been "moved down", so l'Hôpital's rule can be used to evaluate:
as shown in an example above. Then:
Other methods of computing limits
Although l'Hôpital'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.
For example
Some elementary algebraic manipulation, however, yields:
And applying l'Hôpital's rule, we have:
and since the cosine function is continuous for all real numbers, the limit may go "inside" as the argument of cosine
On the other hand, a simple substitution also allows the use of l'Hôpital's rule.
Therefore, as ,
Therefore,
Logical circularity
In some cases it may constitute circular reasoning to use l'Hôpital's rule to evaluate such limits as
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, the argument uses the expected proof to prove itself — i.e. begging the question — and is therefore fallacious (even though the conclusion of the proof happens to be true).
Another popular mistake of this nature is the well-known limit
Just as in the previous example, if one proves the differentiation rule for with the limit, as is often the case in most calculus books, then it is circular to apply l'Hôpital's rule to evaluate the same limit.
External links
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