Who discovered the indeterminate forms and how did they discover them? How did someone come to know that a particular form (fraction, product, sum/difference, exponent) is indeterminate? For example, $\frac{0}{0}$ is an indeterminate form. How did they come to know that when both numerator and denominator approach zero the fraction can be any number (which depends upon their respective rates)?

I’ll be highly grateful if you explain it comprehensively, I’m a high school student.


1 Answer 1


Special cases were handled algebraically even before the "l'Hopital's" rule, which appears in l'Hopital's 1696 transcription of tips on calculus he purchased (literally) from Johann Bernoulli in 1694, see Indeterminate Forms Revisited, by Boas. For example, Descartes's method of finding tangents involved resolving "indeterminate forms" like $0/0$, see Is there a 'lost calculus'? So the phenomenon was known from examples (without a name or special attention paid to it) by the time it was singled out by Bernoulli, and then comprehensively systematized by Euler.

That was done in Euler's textbook Institutionum Calculi Differentialis (1755), chapter 15 of part II. Fortunately, there is an English translation by Bruce. At the beginning Euler explains how $0/0$ come up, why they are "indeterminate", and then gives some tricks for resolving them, including cancellation, "l'Hopital's" rule and logarithmic differentiation. He freely manipulates infinitesimals, and at the end even deduces the famous sum of the Basil series by applying the "l'Hopital's" rule thrice. Here is from the opening, where he shows that arbitrary value is possible with a remarkably simple example:

"If the fraction $\frac{P}{Q}$ were some function $y$ of $x$, the numerator and the denominator of which likewise may vanish on putting a certain value in place of $x$, then in that case the fraction $\frac{P}{Q}$ may arise expressing the value of the function $y=\frac00$; which expression thus may be considered indeterminate, since for each quantity either finite or infinite, or infinitely small it may become equal to, from that evidently in this case the value of $y$ cannot be deduced. Yet meanwhile it is easily seen, because in addition in this case the function $y$ takes a determined value always, whatever may be substituted for $x$, also in this case an indeterminate value of $y$ cannot be possible. This is made clear from this example, if there were $y=\frac{aa-xx}{a-x}$, so that on making $x=a$ certainly there becomes $y=\frac00$. But since with the numerator divided by the denominator it may become $y = a + x$ , it is evident, if there is put $x= a$ to become $y=2a$, thus so that in this case that fraction $\frac00$ may be equivalent to the quantity $2a$."

Although Euler uses the (Latin version of) "indeterminate" he does not call them "indeterminate forms" or introduces handy notations and classifications encountered in modern textbooks. According to Jeff Miller's Earliest Uses, this taxonomic process began in 1840s:

"The term INDETERMINATE FORM is used in French in 1840 in Moigno, abbé (François Napoléon Marie), (1804-1884): Leçons de calcul différentiel et de calcul intégral, rédigées d'après les méthodes et les ouvrages publiés ou inédits de M. A.-L. Cauchy, par M. l'abbé Moigno. Indeterminate Forms is found in English as a chapter title in 1841 in An Elementary Treatise on Curves, Functions, and Forces by Benjamin Pierce. Forms such as $0/0$ are called singular values and singular forms in in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson. In Primary Elements of Algebra for Common Schools and Academies (1866) by Joseph Ray, $0/0$ is called "the symbol of indetermination."


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