Talk:Indeterminate form

message: The form oo^oo is missing from the list.
::How can ∞∞ be considered an indeterminate form?  If f(x) and g(x) both approach ∞ as x approaches a, then f(x)g(x) can only approach one thing, ∞, regardless of which functions f and g are.  So it's not an indeterminate form. Michael Hardy 23:34, 18 December 2005 (UTC)
replies: text: Remember that when one says ∞, one means "positive infinity" (i.e. increasing without bound).  So your example is not the same because it raises "positive infinity" to "negative infinity". --Spoon! 11:33, 31 August 2006 (UTC)

text: What one means depends on context; compare the extended real line to the real projective line. See also this formal definition, which uses the Riemann sphere. —Toby Bartels 22:03, 20 September 2006 (UTC)
thread_title:  Missing Infinity ^ Infinity 

message: User:Michael Hardy, at 15:37, 18 December 2005, wrote in this diff:
: This merger proposal has no merit. Too many readers keep failing to see that "indeterminate form" does NOT mean something that is undefined. Maybe we should hit that point much harder. 
(copied here for future reference by Oleg Alexandrov (talk) 00:16, 19 December 2005 (UTC)
replies: text: To be honest I'm not exactly sure what MH's getting at there. An indeterminate form per se is not undefined, but its value is, and they're pretty much coextensive with the expressions treated at defined and undefined. The latter is just a bad article with a bad title, and I think the real point of the merger was to figure out some way to get rid of it, a project I'd support. --Trovatore 16:58, 13 February 2006 (UTC)
replies: text: No -- they're NOT coextensive, since 1/0 is NOT an indeterminate form.  0/0 is an indeterminate form; 5/0 is not.  That means that if f and g both approach 0 as x approaches something, then f/g could approach anything.  That is not true if f approaches 5 and g approaches 0. Michael Hardy 22:08, 13 February 2006 (UTC)
replies: text: OK, true, 1/0 is not an indeterminate form, but it's not enough by itself to justify a separate article from this one, particularly one with as bad a name as "defined and undefined". And all the indeterminate forms are denotationally undefined (unless you're going to take the 00=1 approach; I'd be against presenting that as the default approach, though it ought to be mentioned). --Trovatore 22:16, 13 February 2006 (UTC)
thread_title:  On merging with [[defined and undefined]] 

message: The "Discussion" section had a problem paragraph (reading "By contrast "1/0" is not an indeterminate form because there is no range of different values that f/g could if f approaches 1 and g approaches 0.").  There was at least a missing word (between could and if).  I attempted to correct that and add more detail to clarify it, but then realized I still don't understand the distinction which is the whole point of the article.
Several times 1/0 is contrasted with 0/0, but the only characterization of this contrast is that there is "no range of different values" without elaboration.  But f/g could have a limit of either +∞ or -∞ depending on whether the function g approaches 0 from the + or - side.  So there seems to be more than one "value" possible among limits approaching 1/0.  Just how is this determinate while 0/0 is not?  Whatever the answer, I think the article needs improvement in explaining this. -R. S. Shaw 00:58, 13 March 2006 (UTC)
thread_title:  Determinate and inderminate discussion 

message: I've changed "The indeterminate forms include [...]" to "The indeterminate forms are [...]" on the assumption that the list is exhaustive.  Previous wording admitted interpretation of list as a subset.
If the given list is not exhaustive, I believe the best compromise is "The indeterminate forms (as described this article) are [...]".  That is to say, whether or not the list is exhaustive, the current wording is needlessly ambiguous. 23:29, 7 March 2007 (UTC)
thread_title: List given in introduction

message: A separate critique is that it seems better to say "are expressed by/using/as" or "are referred to by/as" in place of "are".  It may be confusing to many readers to say such-and-such is 0/0 as it might seem to imply (as we almost always intend when writing such expressions) that 0/0 represents a number rather than a form.  I have not made this change; I don't know what wording best resolves this potential misreading.  Please discuss here.  23:35, 7 March 2007 (UTC)
replies: text: The real problem is that although numerous books use the term "indeterminate form", I have never seen published a definition of one.  Can you think of a textbook that defines them? CMummert · talk 00:25, 8 March 2007 (UTC)
thread_title: List given in introduction (second concern)

message: (This goes back to Carl's remark that "It's more a question of how closely you want the sources to match the phrasing of the article.") I don't think that's the whole issue. While no doubt someone has done it, I don't believe it has ever been standard to give a general, abstract definition of "indeterminate form". If you want to discuss discontinuous funcions, you talk about discontinuous functions; the "indeterminate form" terminology is a fifth wheel when you get to that level of abstraction.
Historically the phrase has been used to give calculus students a way of detecting situations where they have to apply extra care when computing a limit, without bringing up anything as sophisticated as the definition of a continuous function from R2 to R. It's an enumerated collection of situations; no rule is given for extending it. I think that should be the focus of this article. Something like, say, ei∞ could easily be added to the list, but historically, it has not been.
If generalizations can be sourced, they can be added, but they should be clearly marked as generalizations of the notion, not part of the standard interpretation of "indeterminate form". --Trovatore 20:13, 12 June 2007 (UTC)
replies: text: Another example of an expression which could easily be argued to be an indetermination, but corresponds to a clearly unary operation, is arg(0), where 'arg' stands for the complex argument. Although some authors apparently define it as zero by convention, arg(z) can actually approach any real number as z goes to zero, depending on the path followed by z in the complex plane. FilipeS 20:38, 12 June 2007 (UTC)
replies: text: But you're still arguing about the details of the presentation of the general case. My point is that, historically, there is no general case. There are exactly seven indeterminate forms. --Trovatore 20:51, 12 June 2007 (UTC)

text: I'm agreeing with you, if you haven't noticed. As for the number of indeterminate forms, it depends on whether you regard infinity and minus infinity as separate entities, or identify them with each other. FilipeS 20:53, 12 June 2007 (UTC)

text: My advice: don't try to explain. It's fairly obvious why, and I see nothing to be gained from making the notion of 'indeterminate form' strict. If some mathematician somewhere decided to state that arg(0) is indeterminate, what would be wrong with that?... FilipeS 23:52, 12 June 2007 (UTC)
replies: text: Please note I am not in any way trying to make that paragraph a definition, only an explanation. Also there is complete agreement at present about the term "indeterminate form" in every reference I have looked at, and I have looked at quite a few in the process of discussing this article in the last few months. — Carl (CBM · talk) 00:25, 13 June 2007 (UTC)

text: I don't feel that the status of the indeterminate forms as the discontinuities of the standard arithmetical operations on the extended reals is in any way obvious, especially to the lay reader.  I agree that the seven indeterminate forms are pretty much fixed, and would support User:CBM's proposed rewrite. –EdC 21:58, 17 June 2007 (UTC)
replies: text: I'm against getting overly clever at explanations. Note also that if we refer to discontinuities, we have to be careful, as there is no such thing as a function being discontinuous (or continuous) at a point not in its domain. The proper phrasing is that the function does not have (respectively, has) a continuous extension containing that point. And I think that's getting too involved for this article. --Trovatore 22:04, 17 June 2007 (UTC)

text: I agree. Plus, f(x, y)=x/y does not have a continuous extension for y=0, x nonzero either, yet the article claims that x/0 is not an indeterminate form. FilipeS 17:38, 19 June 2007 (UTC)

text: I would say that indeterminate forms are mathematical dead ends. When one shows up in a limit you're trying to compute, that's a sign that you must turn back and take a different route to the calculation. FilipeS 01:03, 22 June 2007 (UTC)
replies: text: Surely her next question would be why they are dead ends? — Carl (CBM · talk) 01:13, 22 June 2007 (UTC)

text: To convince her, I might give a few examples of different functions whose limit at a given point leads to some indeterminate form (say, 0/0), but that when computed in a different way turn out to be different numbers. FilipeS 01:18, 22 June 2007 (UTC)
replies: text: You're beating around the bush.  It's fully reasonable, and expected, for an encyclopedia article on indeterminate forms to do more than give a list, an example of two limits, and no insight whatsoever. Doing so is really a disservice.  I'm planning to remove the paragraph in question - in fact, I'll do it right now - but it should be repalced by something that conveys some intuition about why these forms are indeterminate but others are not. — Carl (CBM · talk) 01:31, 22 June 2007 (UTC)

text: It's not very civil of you to ask a question and then say you don't like the answer. This is your pet article, not mine. You sort out the OR mess you've made of it. Don't expect any help from me. FilipeS 01:33, 22 June 2007 (UTC)
replies: text: I don't agree there was OR in the previous version - I've read a lot of published sources and it was a relatively clear statement of the general principle. If you don't wish to help, I encourage you not to respond to my requests for comment. — Carl (CBM · talk) 01:39, 22 June 2007 (UTC)

text: Funny, then, how you can't seem to be able to come up with any references, or even a coherent definition. If you'd taken any longer to delete that paragraph, I would have done it myself. FilipeS 01:51, 22 June 2007 (UTC)
thread_title: Generality, abstraction, sophistication

message: in the 0^0 form it sayed that if f & g both analytic and g in not fixed zero in the nighberhood of the limit
than f(x)^g(x) is always 1 in that limit which is just false
it is often the case but consider the example:
f(x)=3^x which is positive for all x & g(x)=1/x
both zeroing at negative infinity but f(x)^g(x) is clearly 3
whats up with that ?
:Your function f is not analytic in a neighborhood of infinity (in fact, it's not even continuous at infinity), and in any case the statement talks about neighborhoods of a point on the complex plane.  Infinity is not a point on the complex plane (although it is a point on the Riemann sphere). --Trovatore (talk) 00:50, 25 September 2009 (UTC)
thread_title: false statment that should be deleted/fixed (by BrainScratcher)

message: I didn't see any mention of negative infinity in the article. I only mention this because my Calculus book (Calculus - Early Transcendentals 6th Edition by James Stewart) says that -∞/-∞ is also indeterminate. Also, what about the forms of -∞/∞ and ∞/-∞? Are they all indeterminate forms? Thanks.
BuddhaBubba (talk) 00:03, 20 July 2010 (UTC)
replies: text: Yes they're all indeterminate,they're all just variants of ∞/∞, the signs make no difference in a multiply or divide except to determine the final sign. Dmcq (talk) 09:26, 20 July 2010 (UTC)
thread_title:  negative infinity over negative infinity 

message: Would Tan(90 degrees) be an indeterminate form. The tan wave goes to infinity at 90. Also if you imagine a right angled triangle and imagine the adjacent side getting smaller and smaller theta gets closer and closer to 90 making tan(theta) closer and closer to tan(90) and therefore undefined. To have a right angled triangle with Tan(90) it's adjacent side would have a length of zero and the hypotenuse and opposite sides would be the same length. 90 would be theta, the angle between the adjacent and opposite sides would be 90 and the third angle would be 0. In effect it would be a straight line. However basic trigonometry still works.
For example, if the hypotenuse and opposite had both got a length of 1 then;
Sin(90)=1/1 which is correct.
Cos(90)=0/1 which is correct.
Sin(0)=0/1 which is correct.
Cos(0)=1/1 which is correct.
Tan(0)=0/1 which is correct.
However, Tan(90)=1/0. Which is an indeterminate form. This means that Tan(90) can be converted to 1/0 (and other determinate forms using l'hopitals rule).
If we were trying to work out the length of the hypotenuse and opposite it wouldn't work because we'd get Tan(90)=Hypotenuse/0. This happens because the hypotenuse and opposite could be any length as long as they are the same, the angles would be unchanged.
As well as this Tan(theta)=sin(theta)/cos(theta) so Tan(90)=Sin(90)/Cos(90). Sin(90) is 1 and Cos(90) is 0. Therefore Tan(90) again equals 1/0. 
So is Tan(90) an indeterminate form? It tends towards infinity on the Tan graph. The limit of Tan theta as theta approaches 90 is ±∞. Also are all of the above valid arguments?  — Preceding unsigned comment added by 86.140.32.172 (talk) 14:40, 20 July 2012 (UTC)
replies: text: 1/0 is not regarded as an indeterminate form, as the limits of quotients that lead to it, if they exist, can only be positive or negative infinity, which isn't an infinite number of possible values. Double sharp (talk) 10:06, 26 October 2012 (UTC)
thread_title:  Tan(90 degrees) 

message: The article claims
:If the functions f and g are analytic and f is not identically zero in a neighbourhood of c on the complex plane, then the limit of f(z) g(z) will always be 1.
This strikes me as patently false: it depends on the "operation" of exponentiation that is ill-defined on the complex numbers and cannot be defined as a continuous function in a punctured neighbourhood of (0,0).  Any sensible interpretation will rely on the multivalued complex logarithm, which yields no limit even if one allows limits of multifunctions. Reliance on a single-valued definition requires a "sensible" branch of the exponentiation to be taken, as it seems to me that spiral branches exist that yield any limit that one chooses.  A complex function f(z)g(z) in a bland statement like this without defining the exponentiation or mentioning the mathematically esoteric aspects seems out of place here. I notice that this has survived edits by not-so-clueless editors, so perhaps someone could explain why this claim should remain in the article, or where I've gone wrong? —Quondum 17:04, 22 February 2014 (UTC)
replies: text: There are two quick solutions:  (1) The 0^0 section in exponentiation states it better (i.e. no "complex plane"), so you can copy what it says.  Or (2) delete this line alltogether.  I'm OK with either option.  MvH (talk) 16:02, 26 February 2014 (UTC)
replies: text: I corrected it, and also shortened it a bit.  I also fixed one other issue, it said that the indeterminate form 00 has been discussed for a long time, but that's misleading (the debate was whether 0^0 has a value or not; that's not the same as debating if it is an indeterminate form or not, because there is no standard definition of "indeterminate form").  There were two links on this issue in that section, I moved the second one, in order to combine it with the first.  MvH (talk) 19:56, 27 February 2014 (UTC)
replies: text: Much better, thanks.  Since this is about indeterminate forms, the debate about the value of the expression is IMO irrelevant here.  I might strip that bit out.  —Quondum 23:37, 27 February 2014 (UTC)
replies: text: That's fine (I merged two mentions of the debate into one, but I overlooked that there was a third spot where this was mentioned).  MvH (talk) 14:35, 28 February 2014 (UTC)
thread_title:  Revisiting 0<sup>0</sup> over complex numbers 

message: Not sure where to place this.
http://dubai-computer-services.com/articles/infinities_by_khawar_nehal_19_mar_2014-1.pdf
If you can guide me as to where this can go, then I can place it there.
Regards, 
Khawar Nehal  — Preceding unsigned comment added by 94.203.214.192 (talk) 10:03, 23 March 2014 (UTC)
replies: text: You may wish to read Wikipedia's policies on original research and verifiability, which limit what may be included in articles.  Talk pages such as this are for discussing the content of these articles.  Many people have been fascinated by questions such as these and have spent a lot of time thinking about infinities, and while particular choices may seem to simplify certain problems, this does not help if this creates a logical inconsistency, which will result in paradoxes and fallacies, such as "proofs" that 1 = 2.  The result has essentially been a discovery of when infinity can be treated as a number, and when operations involving zero and infinity are reasonable (see, for example, Real projective line and Riemann sphere; also Transfinite number).  I'd suggest, if you wish to ask about aspects of consistency, the only place on Wikipedia where this might make sense is at Wikipedia:Reference_desk/Mathematics.  —Quondum 16:01, 23 March 2014 (UTC)
thread_title:  A new idea 

message: This article section used to read:
:0∞ also is sometimes incorrectly thought to be indeterminate: 0+∞=0, and 0-∞ is equivalent to 1/0.
Now it reads:
:The limiting form 0∞ is not an indeterminate form. The form 0+∞ has the limiting value 0 for the given individual limits, and the form 0−∞ is equivalent to 1/0.
The changes made were not for the better. A simple and correct sentence was changed into one that is more convoluted, more obscure, and actually somewhat less accurate: algebraic expressions such as 0+∞ or 02 or 01/2 do not have "limiting forms"; functions do. At most, one might say that 0+∞ is a limiting form. And everyone agrees that 0+∞ can be defined as zero, so there's no mathematical sin in plainly writing "0+∞=0". FilipeS (talk) 16:27, 6 August 2014 (UTC)
replies: text: I disagree. The statement is about limiting forms, not about algebraic expressions. As algebraic expressions they are simply nonsense. —Quondum 06:33, 7 August 2014 (UTC)
thread_title:  Undefined forms that are not indeterminate 

message: The unique reference provided here is to an obscure article in an obscure journal by Louis M. Rotando and Henry Korn (January 1977).  Is this really the ultimate reference on indeterminate forms? Tkuvho (talk) 09:50, 11 January 2015 (UTC)
replies: text: Most calculus books simply give the list without a definition.  Membership in the list is determined by historical reasons, not mathematical reasons.  The reason we can only find obscure references for a definition is because as soon as you write down a definition, you'll find that there are other expressions like floor(0) that meet the definition, and that's a situation that most textbooks want to avoid because they don't want to include expressions weren't in the list before. Whether or not floor(0) is an "indeterminate form" depends on who you ask, it's not a concept with an agreed upon definition.   MvH (talk) 15:53, 19 April 2015 (UTC)
replies: text: I think that the added example is not going to serve to do much but confuse, so I don't see that we win any explanatory power by including it. It does not seem to fit the definition in the lead, since no substitution of subexpressions seems applicable at a first glance (it juts looks like an unambiguously nonexistent limit, sucg). But if one retains the example, we must make the fit clearer. My attempt to do so might be to consider the case \lim_{x \to 0} \lfloor -x^2 \rfloor = -1, but \lfloor \lim_{x \to 0} -x^2 \rfloor = 0. In essence, unless we present an unambiguous definition of an indeterminate form, we cannot produce examples or counterexamples without tripping over our own feet. Something about the "feel" of the thing is that it should be something of the form \lim_{t \to c}f(x(t),y(t),\ldots), where the functions x, y, ... can be perturbed continuously within the domain of f to change the (well-defined) limiting value. As you suggest, a good definition is a thorny matter. —Quondum 17:25, 19 April 2015 (UTC)

text: I would like to se a citation for the flor function being considered an indeterminate form. Indeterminate forms refers to arithmetic operators rather than to functions. I know they are functions but the important point is that they are manipulated as operations without checking rules each time. If everything was considered as a function none of this would be necessary - but then we wouldn't be able to do arithmetic simplification without checking the rules for moving things from outside a function call into it.. Dmcq (talk) 17:43, 19 April 2015 (UTC)
replies: text: "Arithmetic function" does not seem to be well-defined. I will discount it being only the elementary operations (+ – × ÷) as then exponentiation would be excluded, but we can consider elementary functions, which would include exponentiation (in its various forms, but only as  if a limit of the exponent is to be involved). On the face of it, this might be a reasonable criterion for classifying something as an indeterminate form, but is this view expressed in any references? I do not see it as being referred to in the article, and cannot only be used for "editorial reasoning". —Quondum 18:46, 19 April 2015 (UTC)

text: Quondum, I searched references for "indeterminate form" some time ago. Most calculus books don't give a definition, and more advanced books generally don't use this term. The books that do give a definition, there were differences, there wasn't a good way to select one or another.  For our definition "if the expression obtained after this substitution does not give enough information to determine the original limit" then all I can say for sure is that it is simply a reformulation of one of the characterizations of a discontinuous function (that's why I added a link to that). Whether or not there are indeterminate forms other than the ones listed, our article is vague on that, could be yes, could be no.  Is this vagueness due to a lack of consensus? The floor(0) example falls in the category "could be yes, could be no".  It might satisfy "the" definition, depending on which one you pick. I placed it after section "Expressions that are not indeterminate forms" because I didn't want to call floor(0) an indeterminate form, since that would seem contrary to historical practice.     PS. Is there a reference for the definition as currently stated? MvH (talk) 19:21, 19 April 2015 (UTC)

text: Regarding "arithmetic function", I think the problem arises from the fact that when this list was made, the word "function" didn't mean the same thing as it does now.  If any definition of "indeterminate form" was made at the time, it would probably be untranslatable to modern math.  Also, even if "arithmetic function" meant precisely what you want it to mean, it would still defeat the educational purpose of this list, because when you deal with limits, you have to be careful with every discontinuous function, not just the ones that appear in a historical list.  Basically, "indeterminate form" is a incomplete list of red flags for limits (the complete list being: discontinuous functions). MvH (talk) 19:35, 19 April 2015 (UTC)
replies: text: Dmcq's argument has some merit: that if we extend a definition to arbitrary discontinuous functions, the whole subject becomes moot (essentially the point you're making), yet we (for pedagogical purposes, not as WP) do want primary examples. I think that "form" might be synonymous with "familiar operation": "elementary function" is too broad. As to "when this list was made", I'd be surprised if the definition of a function was not pretty mature in Cauchy's time. This might suggest changing the focus of the article from assuming that there is a definition and the maths of it to more of a historical commentary on a vague concept, making it clear that it is vague. —Quondum 20:11, 19 April 2015 (UTC)
replies: text: Yes arguments saying that divide is a function and floor is a function therefore why is floor not in the list are treating the subject too abstractly. Divide is not normally considered as a function but as an operator. One would not apply the term 'form' to functions. Overall if the sources are not definite then the article can't be definite, Wikipedia isn't a textbook but an encyclopaedia with some fairly strict content policies. Dmcq (talk) 21:00, 19 April 2015 (UTC)

text: I do not think that \lfloor 0 \rfloor should be considered an indeterminate form unless some reference does so.  In the absence of a claim that such a reference exists, I'm removing the section.  — Arthur Rubin  (talk) 21:38, 19 April 2015 (UTC)
replies: text: I'm happy with removal. But the bigger issue remains: the article pretends to define an indeterminate form, whereas it can strictly only list examples of expressions considered so, and the current definition should be changed into a statement of utility. —Quondum 21:44, 19 April 2015 (UTC)
replies: text: Before you delete the floor(0) example, please consider the issue that it is trying to address. The issue is that the article is vague on what indeterminate forms are:   Are they just the ones that are listed, or are there more?  If there are just the ones that are listed then there is something strange about trying to find a definition.  MvH (talk) 22:30, 19 April 2015 (UTC)
replies: text:  In my experience, the notion of "indeterminate form" is typically presented without any abstract definition at all, just a list, with perhaps a few words about what they have in common, but with no attempt whatsoever to say what a general indeterminate form not on the list might be.  I think that's what we should do as well.  And I don't think we should say anything about floor(0). --Trovatore (talk) 17:19, 20 April 2015 (UTC)
thread_title:  a unique reference? 

message: Now that I've linked Algebraic operation, I notice that three of the listed indeterminate forms (namely those that involve exponentiation) relate to a function that is not an algebraic operation as defined by that article.  It think that it is reasonable to deduce that the definition is flawed, since the cases of exponentiation are clearly historically notable examples. —Quondum 23:46, 19 April 2015 (UTC)
replies: text: Books normally just list the various ones rather than saying anything too explicit. The very first example I get from Google books when I stick in 'indeterminate form' is  which is practically identical to what is said here except they say function instead of expression. And a couple of paragraphs on it just lists the various indeterminate forms. I don't think I've seen any definition that talks about algebraic operators in the modern sense without continuous powers, you're talking about fields which never came into the thinking about calculus and continuous functions. Dmcq (talk) 00:23, 20 April 2015 (UTC)
replies: text: I was talking about what the lead says, and it sounds like you agree that what it says should be changed. The book you link to seems a little inconsistent, in the sense that it defines an indeterminate form as a function that is undefined at a point of its domain (but, implicitly for which the limit may exist).  So, we can look at rewording the lead to eliminate the reference to an algebraic operation. —Quondum 02:42, 20 April 2015 (UTC)

text: Okay, done. I hope that I have adequately addressed this in the lead. —Quondum 03:03, 20 April 2015 (UTC)
replies: text: Seems fine to me. Dmcq (talk) 07:45, 20 April 2015 (UTC)
thread_title:  Algebraic operations? 

message: 0/0 is indeterminate, russian word equivalent sense will be "неоднозначно". 
10/0 is undefined, russian word equivalent sense will be "неопределено".  188.208.126.82 (talk) 14:11, 9 June 2023 (UTC)
thread_title:  Russian translation equivalent sense for words undefined and indeterminate