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L'Hôpital's rule is (essentially) a trick for finding limits of various indeterminant forms. It is probably due to Jean Bernoulli on whose lectures l'Hôpital's calculus textbook was based.

The easiest form to prove is one which assumes that both f and g are continuous and differentiable functions near x=a and both have value 0 at a. We seek

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Standard techniques would give the form 0/0, which is indeterminant since examples can be found easily where the limit is 0, a number, or infinite (try letting f and g be powers and a=0 to come up with your own examples).

The intuition behind l'Hôpital's rule is that finding

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is the same as finding

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Now the mean value theorem applied to f and to g would say this is the same as

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where a' and a'' are between x and a. The factors of (x-a) could then be canceled. If only a' and a'' were the same this would give l'Hôpital's rule. Since the mean value thoerem doesn't tell us that they are the same, we need a form of MVT which works with two functions instead of one.

Theorem53

PROOF:

Apply Rolle's theorem to the function

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This function h is continuous on [a,b] because both f and g are, and is differentiable on (a,b) with

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At the end points we get

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By Rolle's theorem there is a c in (a,b) with h'(c)=0. This gives the needed point.

Cor63

PROOF:

As before we think of

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where tex2html_wrap_inline199 is given by the Cauchy Mean Value theorem. Now since c is between x and a we know that as tex2html_wrap_inline207 , we also have tex2html_wrap_inline209 . Thus

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This particular form of l'Hôpital's rule is not the most useful one. That is the tex2html_wrap_inline213 form for limits as tex2html_wrap_inline215 . The major use of that is to show that asymptotically tex2html_wrap_inline217 grows slower than any power and tex2html_wrap_inline219 grows faster than any power. The other major use of this form is to show that

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a formula used for approximating compound interest by continuous compounding, one of the major uses of the exponential.

There are several standard pitfalls in using l'Hôpital's rule. The most obvious involve trying to use it on limits not of the form 0/0 or tex2html_wrap_inline213 . The limit

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which forms the basis for the derivation of the derivative of sin(x) needs a geometric argument based on the definition of radian measure and the trig functions as coordinates, rather than using l'Hôpital because the limit is needed to find the derivative of the tex2html_wrap_inline229 function, so that derivative cannot be used in finding the limit.



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Sat Mar 13 10:08:30 CST 1999
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