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In an exam we have

Question (5points) find $\lim_{x\to\infty}(x-\sqrt{x})$.

A student answered: $\lim_{x\to\infty}(x-\sqrt{x}) =\lim \sqrt{x}(\sqrt{x}-1)=\infty \cdot\infty=\infty$.

My question is: how many point you will give the student? A professor in our department says it should be zero or one out of five, arguing that the student does not understand what are limits for and treat them as real numbers.

I think 4/5 is fair. Because he almost did everything except that he did not mention that the product of two infinite limits is infinite.

Edit: I am talking about the first course in the university which is calculus.

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    $\begingroup$ To be fair that's common notation and I don't think it implies misunderstanding of the concept of infinity. It's however false that $\sqrt{x}(x-1)=x-\sqrt{x}$. $\endgroup$ – F.Webber Dec 27 '17 at 20:58
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    $\begingroup$ IMHO: What did you teach them? If you were formal and spent hours explaining to students that limits are not real numbers, that operating on infinities is dubious at best and then taught them proper techniques for evaluating such limits than I think the student deserves a 0 or 1 out of 5. On the other hand if you gave them some practice tools for figuring out a limit as "what does the expression go to," did a couple of homework problems where the expectation was for them to write the correct answer (and not derive it correctly) then I think you should give them a 4/5. $\endgroup$ – Nate Bade Dec 27 '17 at 23:54
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    $\begingroup$ I think I would want to know what a perfect answer should look like to you in order to adequately discuss the question. $\endgroup$ – rnrstopstraffic Dec 28 '17 at 0:38
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    $\begingroup$ I'd give him full credit. It said to find the limit, and he found it. His justification that $\infty\cdot \infty = \infty$ is just as good as "if $f(x)\underset{x\to\infty}{\longrightarrow}\infty$ and $g(x)\underset{x\to\infty}{\longrightarrow}\infty$, then $f(x)g(x)\underset{x\to\infty}{\longrightarrow}\infty$" as can be expected from a student who (most likely) has never taken a proofs class. If the professor wants to punish people for using hand wavy notation, I sure hope he's teaching them how to write the epsilon delta proofs he's expecting. $\endgroup$ – Alexander Gruber Dec 28 '17 at 4:04
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    $\begingroup$ @MuathKaraki In fact $\infty\cdot\infty=\infty$ is perfectly valid with the proper definition of $\infty$. It even shows a good understanding of the concept of limit, or at least, it does not show any lack thereof. $\endgroup$ – Miguel Dec 28 '17 at 9:51

16 Answers 16

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The student changed something which was indeterminate ($\infty-\infty$) into something which was not ($\infty\cdot \infty$). How does that not merit a perfect score? Changing indeterminate expressions into determinant ones is, generally speaking, the point.

If the professor had some other solution in mind, then they made a mistake. They should have chosen a question without an easy out. It happens; I've certainly had students provide solutions which short-cut what I had planned. That, however, does not make the answer incorrect.

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    $\begingroup$ The student does not seem to have written $\infty - \infty$ (at least as the OP reported things). Rather the student replaced $\sqrt{x} - 1$ with $\infty$ directly. $\endgroup$ – Dan Fox Dec 28 '17 at 9:30
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    $\begingroup$ @DanFox Why would they have written $\infty-\infty$? $\endgroup$ – Adam Dec 28 '17 at 15:47
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    $\begingroup$ More to the point, I normally solve this with a decision ladder, but if the student doesn't have the ladder yet I can't think of a better way to solve it. I have a hunch the professor would have hated "∞ by inspection" or "∞ by ladder" for my answer. $\endgroup$ – Joshua Dec 29 '17 at 2:21
  • $\begingroup$ @Adam: Your answer attributes to the student writing $\infty - \infty$. That would occur from replacing each $x$ in $x - \sqrt{x}$ by $\infty$, and would be a common, incorrect, line of reasoning. $\endgroup$ – Dan Fox Dec 29 '17 at 15:29
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    $\begingroup$ @DanFox I believe that you are misreading my statement. A student who has mastered precalc should automatically know that positive powers of $x$ tend to $\infty$ as $x\to \infty$. Thus, they should think "Ah, an indeterminate expression of the form $\infty-\infty$, let's see what I can do with that..." in their head. There is no substitution $x=\infty$. If there had been, you would have seen the tell-tale $\infty(\sqrt{\infty}-1)$ in the penultimate expression. $\endgroup$ – Adam Dec 29 '17 at 15:59
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If this is calc I, that deserves a 5/5. If this is analysis, it depends on what you taught them. Don't you set up a grading rubric ahead of time? What do the 5 point answers look like? What do other not-so-great answers look like?

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    $\begingroup$ I agree, but I would add 'if this is calc I and you have not specified to support formally every step...' $\endgroup$ – Miguel Dec 28 '17 at 9:55
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    $\begingroup$ As @gnasher729 said in a comment above, "i'm absolutely flabbergasted with the whole discussion here". In class I would typically write this as (big)(big) = big or $(\rightarrow \infty)(\rightarrow \infty) \rightarrow \infty,$ saying that when $x$ is really big, then each of the factors is big and thus the product is big. I rarely (maybe never?) in a beginning calculus class wrote things like $\infty \cdot \infty = \infty,$ (continued) $\endgroup$ – Dave L Renfro Dec 29 '17 at 8:15
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    $\begingroup$ but I also wouldn't take off for this on a test unless I had given very explicit instructions about what is acceptable and what is not, which at this level is very difficult to do without losing the forest for the trees (and thus I didn't). If instead the student had simply written $\infty - \infty = \infty,$ then on the student's test my comments and point deductions would be for the fact that big numbers subtracted from big numbers do not always produce big numbers, and not for symbolic manipulations with the $\infty$ symbol. $\endgroup$ – Dave L Renfro Dec 29 '17 at 8:15
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    $\begingroup$ For what it's worth, the first two "General Remarks" at the end of this 2-page handout of mine is how I dealt with this later, in a second semester U.S. college calculus class when talking about limits in the sequences and series material (this comes after integration techniques and their applications), and so is pitched at a higher level than it would be in an introductory treatment of limits in a precalculus course or at the beginning of a first semester calculus course. (continued) $\endgroup$ – Dave L Renfro Dec 31 '17 at 9:43
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    $\begingroup$ Incidentally, the tone of those remarks is a bit harsh. I originally wrote this in Spring 1997 while teaching a third semester calculus course to some very strong high school students (here, where I taught a few years in the 1990s) and I was a bit upset at the amount of effort I was seeing (this was at the beginning of the course). I was rather busy when I wanted to use the same handout later, so I just changed the title to where I was later teaching at and told students to not take the comments too personally. $\endgroup$ – Dave L Renfro Dec 31 '17 at 9:51
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Instead of arguing with other people's answers in the comments I thought it might be more productive to present my own point of view. I find myself completely unable to understand why anyone would take off points for this student's answer.

Just to be clear, this isn't because I'm being somehow lax or generous as a grader. My opinion is that this is a model solution to the problem, written clearly and well, and I can imagine writing exactly what this student wrote as part of homework solution or exam solution that I distribute to a class. In the context of Calculus I, it's also how I would do this problem on the board during class if a student asked me about it.

On the Status of Infinity

Some of the other calculus teachers here have mentioned that they teach their students that "infinity isn't a number".

I find this statement very strange, and I suppose that my position is that infinity is a number. It certainly isn't a real number, since it's not included in the usual real number system. But neither is the imaginary unit $i$, and I don't think many people would argue that $i$ isn't a number. The number $i$ is included in the system of complex numbers, and the number $\infty$ is included in the system of extended real numbers, which is the set $\mathbb{R}\cup\{-\infty,\infty\}$. I don't see the difference.

Of course, there's no standard definition of "number" in mathematics, so there's no objective truth either way. This is part of why it strikes me as so odd that a teacher would say that "$\infty$ isn't a number".

It's possible that what they mean is that "you can't do arithmetic with $\infty$". But of course you can do arithmetic with $\infty$. For example, $$ \infty + \infty = \infty,\qquad \infty \cdot \infty = \infty,\qquad\text{and}\qquad 3\cdot \infty = \infty. $$ These definitions are absolutely standard in mathematics, and I would feel free to use them in a conference talk or journal article without comment. I would hope that most calculus students would know how to do basic arithmetic with $\infty$ by the end of a first calculus course, but apparently this varies by instructor.

There are also arithmetic operations involving $\infty$ that are undefined, such as $$ \infty - \infty,\qquad \frac{\infty}{\infty},\qquad\text{and}\qquad 0\cdot\infty. $$ The last is sometimes defined to be zero (e.g. in the theory of Lebesgue integration), but in the context of calculus it's better to leave it undefined.

As far as I know, all of this is completely standard, and in my experience arithmetic involving $\infty$ and $-\infty$ is commonly used by mathematicians without further explanation or comment. I've seen lots of examples of this, but to cite a specific one it's certainly the case that Rudin's Real & Complex Analysis textbook (an extremely standard choice for a graduate analysis course) uses the extended real number system throughout.

On the Student's Answer

The student's answer depends primarily on the following theorem

Theorem. Let $f\colon \mathbb{R}\to\mathbb{R}$ and $g\colon\mathbb{R}\to\mathbb{R}$ be functions, and let $a\in [-\infty,\infty]$. If $$ \lim_{x\to a} f(x) = L\qquad\text{and}\qquad \lim_{x\to a} g(x) = M $$ for some $L,M\in[-\infty,\infty]$ and the product $LM$ is defined, then $$ \lim_{x\to a} f(x)\,g(x) = LM. $$

This is a well-known and standard theorem in analysis. In the context of this theorem, the student's work constitutes a perfectly good proof of the fact that $$ \lim_{x\to\infty} \bigl(x-\sqrt{x}\bigr) = \infty. $$ It is no more or less correct than something like $$ \lim_{x\to 0} \frac{x\sin x + 2 \sin x}{x} = \lim_{x\to 0} \,\bigl(x+2\bigr)\!\left(\frac{\sin x}{x}\right) = (2)(1) = 2. $$ I don't see why this proof would require any more explanation or rigor, in either a calculus or real analysis course, and I feel the same way about the student's proof. I suppose it might be reasonable for an analysis professor to always require students to cite the theorems that they are using, as opposed to using theorems implicitly as part of a calculation. I certainly don't think this would be a reasonable requirement for student answers in a calculus course.

Should we teach arithmetic with infinity to calculus students?

I do, and I would certainly hope that most other calculus instructors do as well. Dealing with the concept of infinity is a major theme of calculus, and the rules for arithmetic involving infinity ultimately derive from the idea of a limit. How does it help to avoid talking about this?

Actually, it seems to me that it would be difficult to cover the idea of an "indeterminate form" without covering this material. I guess at least some of the teachers here manage to avoid saying that "infinity plus infinity equals infinity" by always saying "the sum of two quantities that are both approaching infinity again approaches infinity", but what's the purpose of being so obtuse? If there's a simple way to say something, just say it that way.

And in any case, the reality is that you can do arithmetic with infinity. Saying that $\infty+\infty$ is undefined or indeed anything other than $\infty$ is just wrong, both at an intuitive level and from the point of view of standard notation and terminology. Students will figure out that it's true on their own, and will try to guess what other arithmetic rules you're not telling them. If you tell students that $\infty + \infty$ isn't $\infty$, you lose your credibility, and they won't believe you later when you tell them that $\infty - \infty$ isn't $0$.

Okay, but should we mark the student wrong?

Even if you don't talk about arithmetic involving infinity in your calculus class, the fact remains that it is absolutely standard mathematical notation. Students often seek help from mathematics tutors, other math professors, online videos, and so forth, and any one of those sources might be teaching your students about how to use infinity in this fashion. Can you really justify deducting points from students who don't write their mathematics the way that you want it written? I feel like one of the most basic principles of grading is that correct answers should receive full credit, unless the answer explicitly violates the instructions for the question. This student's answer is completely correct, and in my opinion giving it anything less than 5/5 is just arbitrary and unfair.

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    $\begingroup$ Well argued. Unless I had given specific instructions to the contrary for some reason, I think I would do this. +1 $\endgroup$ – G Tony Jacobs Dec 31 '17 at 0:52
  • $\begingroup$ This argument basically boils down to "infinity is too a number". I would like to see an example of a freshman calculus text that does define the extended reals and its operations. The only example I can lay my hand on (Stein and Barcellos, Calculus and Analytic Geometry, Sec. 2.4) explicitly says, "$\infty$ is not a number". So based on my one data point, it looks like this answer is predicated on directly contradicting one's freshman calculus text. Which would make me very uneasy. $\endgroup$ – Daniel R. Collins Dec 31 '17 at 1:28
  • $\begingroup$ Likewise, OpenStax Calculus Vol. 1 does not have the phrase "extended real" anywhere in the text, and says of infinite limits, "We are not asserting that a limit exists." (Sec. 2.2) $\endgroup$ – Daniel R. Collins Dec 31 '17 at 1:42
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    $\begingroup$ @DanielR.Collins The problem is that generally a real number has not been defined, a limit has not been defined, etc. These things happen in a real analysis class. At this level, the justification really is at the level of "big number times a big number is still big". Penalizing someone for writing $\infty \cdot \infty = \infty$ in this context seems weird. When expected rigor is extremely variable, and is adjusted to the whims of the instructor, it seems like rigor is something we do to keep an authority happy, instead of a way to clarify and communicate with others. $\endgroup$ – Steven Gubkin Dec 31 '17 at 19:26
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    $\begingroup$ @DanielR.Collins Many such books also include sections on hyperbolic trig functions. How many courses expect students to know the definitions of these functions? They include "proofs" of all of the major results. How often is understanding of these proofs tested? A curriculum is not determined by the book at all. It is decided by the instructor. $\endgroup$ – Steven Gubkin Jan 1 '18 at 18:08
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When I taught math, I used to follow the principle "do no harm", or "innocent until proven guilty". In other words:

  • Maybe here we deal with a student who is treating infinity as a number. ("Guilty")
  • Maybe we are dealing with someone who knows exactly what they are doing and have found a good shortcut ("Innocent").

Because I cannot tell, based on the answer, I would mark it as 5/5 and be careful to next time ask a better question, which would hopefully let me better distinguish the "guilty" from "innocent".

That's me. YMMV.

Note: In this case, my guess is that the student is "innocent". A "guilty" student is, I guess, far more likely to write that the solution is $\infty-\infty=0$, or $\infty-\infty=\text {indeterminate} $ or something like that.

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    $\begingroup$ I don't like "innocent until proven guilty" applied to the classroom. To see the flaw in this philosophy, take it to the extreme: A student who skips all the exams hasn't proven he knows nothing, so you have to give him an A; he has submitted no wrong work. If this were my student, by giving such an answer, he would have proven that he hadn't been attending class. So "guilty" anyway. $\endgroup$ – B. Goddard Dec 27 '17 at 22:04
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    $\begingroup$ Fair enough. I wanted to express what my aim always was - to avoid being one of those teachers who have only two ways of solving problems: "my way" and "the wrong way", even at the cost of some students sneaking through when they didn't deserve it. I am not saying I myself was particularly consistent at sticking to my own philosophy, either... $\endgroup$ – user8734617 Dec 27 '17 at 22:13
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    $\begingroup$ Honestly, with the exception of finals, I almost never give the student the benefit of the doubt (in your language, I grade "guilty until proven innocent"). I have two reasons for this: (1) I think that an important part of mathematics is clearly communicating one's thoughts, and (2) I would rather deduct points while grading and have a strong student prove that they actually did know what they were doing in office hours (where I can refund the points) than award the points to a weak student who actually had no idea. That said, the OP's student deserved (nearly?) full marks IMO. $\endgroup$ – Xander Henderson Dec 28 '17 at 21:19
  • $\begingroup$ This is nearly the opposite reasoning than what we are taught as high school teachers: Assume students are "innocent of all learning until proven guilty." However, in high school, there is much more room for formative assessment. $\endgroup$ – Opal E Dec 30 '17 at 23:32
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Calculus classes are taught at an 18th century standard of rigor, and analysis classes at a 20th century standard of rigor. It doesn't make much sense to try to invent some arbitrary combination of the two. So if you aren't expecting proofs written in sentences with all proofs eventually going back to epsilon-delta definitions, then you should accept anything that Euler would have written. Would Euler have written this? Of course he would have! Full points.

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  • $\begingroup$ Euler didn't use epsilon deltas...just sayin $\endgroup$ – jfkoehler Jan 11 '18 at 20:44
  • $\begingroup$ And neither do calculus students, that’s exactly my point. $\endgroup$ – Noah Snyder Jan 11 '18 at 21:26
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The answer seems ok to me, in that it shows that the student understands what the limiting behavior of $x - \sqrt{x}$ is as $x \to \infty$. If the questioner wants to see a formal justification of that, then the word "find" should be replaced by some more precise indication.

The interpretive problem is that that the exercise could be better posed. A better way to ask the same question is simply to ask: what is the limiting behavior of $x - \sqrt{x}$ as $x \to \infty$?

In the classical sense the limit $\lim_{x \to \infty}x^{2}$ does not exist. What is meant by writing $\lim_{x \to \infty}x^{2} = \infty$ is different than what is meant by $\lim_{x \to 2}x^{2} = 4$, because $\infty$ is not a real number. The expression $\lim_{x \to \infty}f(x) = \infty$ is a shorthand that means that $f(x)$ is eventually (on the positive real line) larger than any positive real number. It is used to distinguish the behavior of $\log{x}$ and $\cos{x}$ as $x \to \infty$. Neither has a limit as $x \to \infty$ in the usual sense, but their limiting behaviors are different, in the sense that one function grows unboundedly while the other is oscillatory, although bounded. This can be indicated by assigning the putative "value" $\infty$ to $\lim_{x \to \infty}\log{x}$ and declaring that $\lim_{x \to \infty} \cos{x}$ "does not exist", even though $\infty$ to $\lim_{x \to \infty}\log{x}$ also does not exist in the usual sense (it cannot be assigned a value in $\mathbb{R}$). So one is using the same notation to indicate a limit that exists in the usual sense and a limit that, although it does not exist in the usual sense, can be given sense in that the limiting behavior of the argument function has a well-defined character ... This is akin to the unfortunate and sloppy conflation of integrals and primitives that one often encounters in first calculus courses; apparently identical notation is used to indicate operationally and/or conceptually distinct entities.

Because the relevant distinction is a subtle for students, it seems to me a pedagogical error to treat $\lim_{x \to \infty}x^2 = \infty$ and $\lim_{x \to 2}x^{2} = 4$ on the same level. Their meanings and interpretations are different. I would try to avoid writing the former expression, or would make a serious effort to explain that it is formal notation indicating something different than the well-defined notation in the latter expression (one encounters similar issues when treating integrals with infinite limits or integrals of unbounded functions).

When one asks "find $\lim_{x\to \infty}f(x)$" it is implicit in the question being well posed that $f(x)$ has a well-defined limiting behavior that can be summarized by a single notational expression (so is not oscillatory), but "find" is really the wrong word.

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    $\begingroup$ "The expression limx→∞f(x)=∞limx→∞f(x)=∞ is a shorthand that means that f(x)f(x) is not bounded on the positive real line." No, a limit of infinity is different from being unbounded. x cos(x) is unbounded, but does not have a limit of infinity. Limit of infinity means that for every M, there exists m such that x>m -> f(x) >M. Equivalently, the limit of 1/f(x) is zero and there is some point after which f is never negative. $\endgroup$ – Acccumulation Dec 28 '17 at 22:47
  • $\begingroup$ @Acccumulation: thank you. The inaccuracy has been corrected now. $\endgroup$ – Dan Fox Dec 29 '17 at 15:32
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It really depends on how this was taught in the course. When working with limits it is useful to define the extended real number system $\overline{\mathbb{R}}:=\mathbb{R}\ \cup \{\infty,-\infty\}$. We call $U$ a neighbourhood of $\infty$ ($-\infty$) if there exists a $K>0$ such that $(K,\infty) \subseteq U$ ( $(-\infty, -K)\subseteq U$ ). Now some sequences which diverge in $\mathbb{R}$ converge in $\overline{\mathbb{R}}$, namely those which "diverge to infinity" (although not all properties of convergent sequences hold). Then it is possible to define addition and multiplication on $\overline{\mathbb{R}}$ by looking at the addition and multiplication of the limits of sequences which converge to $\pm\infty$. For example, if $(x_n)_{n\in\mathbb{N}}\subseteq\mathbb{R}$ and $(y_n)_{n\in\mathbb{N}}\subseteq\mathbb{R}$ both tend to infinity for $n\rightarrow \infty$, then $(x_n+y_n)_{n\in\mathbb{N}}$ goes to $\infty$ as well, thus we define $\infty + \infty = \infty$. In a similar fashion one obtains $\infty \cdot \infty = \infty$. Note that for instance $0 \cdot \infty$ has to be left undefined because $\lim_{n\rightarrow\infty}\frac {1}{n}\cdot n =1$, but $\lim_{n\rightarrow\infty} 0 \cdot n = 0$.

In this sense of the real extended number system $\infty$, contrary to some other answers here, becomes a number (although $\overline{\mathbb{R}}$ doesn't form a field) and it is perfectly valid to argue like the student did.

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  • $\begingroup$ This is technically correct, but in a calculus course it would only serve to confuse the students. $\endgroup$ – Martin Argerami Dec 30 '17 at 12:01
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    $\begingroup$ @MartinArgerami I don't know what the equivalent of a Calculus course is over here in Germany, but I am currently taking a first semester course in "Analysis" and that's how it was taught in the course. $\endgroup$ – Jannik Pitt Dec 30 '17 at 12:53
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    $\begingroup$ @MartinArgerami I regularly teach calculus this way and I don't understand what would be confusing. Since it's true that $\infty \cdot \infty = \infty$, why would it be less confusing to ignore or deny this fact than to confirm it? $\endgroup$ – Jim Belk Dec 30 '17 at 14:56
  • $\begingroup$ @Jim Belk: My understanding of Martin Argerami's comment is that introducing the extended real line in a beginning calculus course is likely to confuse students. I agree this would be the case for a typical U.S. first year course, although apparently not for a first year German course. That said, I'm sure there are several honors level calculus texts (such as the books listed here) that introduce and use extended real line arithmetic. But this would only apply to a small percentage of U.S. calculus students. $\endgroup$ – Dave L Renfro Dec 30 '17 at 21:22
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    $\begingroup$ @Jim Belk: Looking at my comment a day later, I think at the beginning of my comment it would better convey what I meant if instead I had said "is that formally introducing the extended real line in a beginning calculus course", where by "formally" I mean talking about neighborhoods of infinity and such, rather than the observation that the product of two things approaching infinity will approach infinity, which I think is perfectly fine. $\endgroup$ – Dave L Renfro Dec 31 '17 at 9:34
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I will add yet another answer saying that this deserves full score, because I feel that two cents are still missing.

  1. The student obviously knows what he does. Do you want to punish such a student? Do you really think it's correct to deduct score points for not mentioning things such as "limit of a product is a product of limits given these limits exist and the product is well defined"? In my opinion, these things should not be sought for while testing practical computations.

  2. If you want to test knowledge of calculus related theorems, there are much much better ways. And testing practical calculus should be separated from testing theoretical knowledge.

Moreover, if you want to test, in practical calculus, things that depend on precise knowledge of the related theorems, you should propose corresponding problems, whose solution without this knowledge is impossible.

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I understand starting a discussion violates etiquette, but I cannot resist:

What was the desired solution?

Was it:

$$ \begin{align*} \lim_{x \to \infty} x - \sqrt{x} &= \lim_{x \to \infty} (x - \sqrt{x}) \frac{x+\sqrt{x}}{x+\sqrt{x}}\\ &=\lim_{x \to \infty} \frac{x^2-x}{x+\sqrt{x}}\\ &= \infty \textrm{ by leading terms theorem} \end{align*} $$

or something like this?

The students solution is very clever. What would they have had to say to get full points for their solution? Maybe quote some theorem that if $\displaystyle \lim_{x \to \infty} f(x)$ and $\displaystyle \lim_{x \to \infty} g(x)$ both tend to infinity, then their product does as well?

In general, what do you expect students to do when confronted with a ``determinate form'', such as $\frac{0}{\infty}$ or $\infty \cdot \infty$ or $0^\infty$?

I am genuinely curious what the OP (and others who agree with him) would like to see.

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    $\begingroup$ I'm curious about the intended solution as well, though this seems like a plausible guess. I tend to think that the student's answer demonstrates a better understanding of limits than this one does. $\endgroup$ – Jim Belk Dec 31 '17 at 18:48
  • $\begingroup$ +1 I was thinking this myself when I was reading over the question and answers (for the first time) yesterday, but after spending a bit of time writing several comments yesterday, I decided I'd spent enough time here. (I've been very busy with contract work at home the past few days, and felt I'd "played around enough".) Then I came back this morning and spent time writing some more comments . . . (Not to mention being here now!) $\endgroup$ – Dave L Renfro Dec 31 '17 at 20:14
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    $\begingroup$ I imagine the intended solution is something like this: “$\displaystyle \lim_{x\to\infty}(x-\sqrt{x}) =\lim_{x\to\infty} \sqrt{x}(\sqrt{x}-1)$ and as $\displaystyle \lim_{x\to\infty} \sqrt{x} = \infty$ and $\displaystyle \lim_{x\to\infty} (\sqrt{x} - 1) = \infty$ and the product of two infinite limits is infinite, the original limit is also $\infty$.” Or something like that, which does not involve writing down “$\infty \cdot \infty$”. $\endgroup$ – ShreevatsaR Jan 1 '18 at 19:46
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    $\begingroup$ I can see the argument for 4/5 if the requirement of writing sentences and citing theorems is applied consistently. For example, when calculating $\lim_{x \rightarrow 0} x+2$ you can’t just break it up as a sum of two limits because you needn’t to know each limit exists before applying the theorem. So first calculate each limit separately and only after that say you’re applying the theorem to get the sum of the separate limits. $\endgroup$ – Noah Snyder Jan 1 '18 at 19:56
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    $\begingroup$ @ShreevatsaR Seems completely arbitrary. Was it okay for you to write $\lim_{x \to \infty} \sqrt{x} = \infty$ when $\infty$ is not a number and the LHS does not exist? $\endgroup$ – Steven Gubkin Jan 1 '18 at 19:56
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I’d give 4/5, i.e., a B grade.

A generous reading has the student mentally glossing over some of the finer points of limit work. Does the student know what’s they’re glossing over? From their work, we can’t tell.

Part of what we teach in math is notation, and part of what we teach is attention to detail. These parts are as important as just “knowing math”. Why don’t I then suggest 2/5, or something closer to a third? Well, I say the first 2/5 is for making an honest attempt, and for the valid steps they did write down. However, they should know that this is a shorthand, and what it's short for.


EDIT: I'm leaving my previous answer up, for accountability, but I've been persuaded by the answers saying the student should receive full credit. If there's anything wrong with their thinking, a different question should be used to determine that.

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    $\begingroup$ As with the other negative answers, I'm utterly puzzled by your complaint. What's being glossed over here? Is the problem that they're not citing the theorem that the limit of a product is the product of the limits? I'm a working mathematician and the student's answer is basically exactly what I would write, so I don't understand why "this is not how math is written". $\endgroup$ – Jim Belk Dec 30 '17 at 14:44
  • $\begingroup$ So you're going around downvoting everyone's answers that you have questions about? Okay... I thought my answer was pretty positive. The professor cited in the question would have given 1/5. As for what's being glossed over, infinity is not a real number. If the students have been taught the extended reals, and which operations are permitted with infinite elements thereof, then that's a different matter, but the OP did not present that context. I would just hope the student knows that $\infty\times\infty=\infty$ is shorthand for something else. $\endgroup$ – G Tony Jacobs Dec 30 '17 at 16:43
  • $\begingroup$ In particular, when we write such a multiplication statement, what we really mean is, "if, as $x\to \infty$, we have $\lim f(x)=\infty$ and $\lim g(x)=\infty$, then we have $\lim f(x)\cdot g(x)=\infty$." There is a family of such theorems, and I teach them, and I teach the shorthands. The OP did not make it clear that any such context was established. I've seen too many of the students who write $\infty\times\infty=\infty$ also write $\frac00 = 0$. It makes me want to be sure they understand the underlying mathematics before leaning on the shorthands. On reflection, I'll modify my answer. $\endgroup$ – G Tony Jacobs Dec 30 '17 at 16:50
  • $\begingroup$ @JimBelk, are you less upset with my modified answer? $\endgroup$ – G Tony Jacobs Dec 30 '17 at 16:52
  • $\begingroup$ I think the new version is a lot more reasonable, and I have removed my downvote. I still don't understand exactly what you mean by "shorthand" -- arithmetic using infinity certainly does make calculations shorter sometimes, but basically all definitions and notations in mathematics are shorthands or tricks for dealing with more complicated things. If a student computes a derivative using the product rule, do you take off points because they glossed over the fact that a derivative is the limit of a difference quotient? $\endgroup$ – Jim Belk Dec 30 '17 at 17:00
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I think context is everything here. When I teach calculus, one of my stock phrases is "infinity is shorthand for a limit." This is an elementary (Calc I) limit, so at this level, the whole point is whether the student understands the background rigor. Here, he is working the problem backwards, making something less rigorous when I told him to make it more rigorous. I would accept the above calculation in a DE course, but in Calc I, probably 0 points out of 5.

I don't really care if the answer is right or if it makes sense or if a "real" scientist/engineer would work it that way. I need to know what's going on in your head, and this answer short-circuits this. It may as well be a multiple-choice question, otherwise.

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  • $\begingroup$ What is DE course? $\endgroup$ – Muath Karaki Dec 27 '17 at 21:26
  • $\begingroup$ Differential Equations. $\endgroup$ – B. Goddard Dec 27 '17 at 21:38
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    $\begingroup$ "I don't really care if the answer is right". From my point of view, if you want to emphasize "rigor", then you should not have posed this question in the first place, or pose it differently. "Rigor" is poorly defined anyway - what do you mean that the student has to do it "rigorously"? Does that mean s/he has to start from defining the real numbers? The square root? The definition of the limit? You may as well tell the student to guess what's in your head. $\endgroup$ – Project Book Dec 27 '17 at 22:44
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    $\begingroup$ @ProjectBook By "rigor" here, I mean that I want them to explain to me in the same way they would explain to another, struggling student. Getting the right answer by means of shortcut or intuition doesn't show mastery of the skills I'm trying to teach. The problem above isn't mine, (I don't think it's a very good exam problem) but at the east I would have insisted that the product of two limits be written. $\endgroup$ – B. Goddard Dec 28 '17 at 1:49
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    $\begingroup$ Like I said, that is too arbitrary a criteria, like for example if the question is "Find $f'(x)$ where $f(x) = \log(\cos(x)), x \in (0,\pi/4)$, then a reasonably trained student can write out immediately what the answer is, whereas if that student has to "explain to another struggling student", he would have enormous trouble. Where does he even start? This sort of question shouldn't have place in exams where mathematical rigour is what is being examined rather than problem solving. $\endgroup$ – Project Book Dec 28 '17 at 2:35
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Sometimes educators like to think about test questions as measuring specific things. What are you trying to assess with the question? Depending on what you’re trying to get, you may want to give full credit and some comment about being careful with statements like

$\infty \times \infty = \infty$

If you want this to test the knowledge of symbolics and operations with infinity I’d give him zero, although I would feel bad about it.

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    $\begingroup$ I just don't get it. Infinity multiplied by infinity is equal to infinity! Why on earth would you teach a student to be careful about doing something that's completely correct? $\endgroup$ – Jim Belk Dec 30 '17 at 14:40
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    $\begingroup$ Because in general you don’t want them to think they operate with infinity like it is a number. This is why the calculus text contains problems with operations with infinity. You’re right, that should’ve been an equals sign, sorry, I’ve fixed it. $\endgroup$ – jfkoehler Dec 30 '17 at 17:18
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There are two mottoes I recite whenever I feel doubt in grading. They are:

  • "If it's not wrong, then it's right."
  • "Was I explicit in what I wanted them to do?"

I would give them a 5/5, if I weren't explicit. I would give them a 4/5, if I said justify any limit rules. I would give a 0/5 if I explicitly said I wanted a $\delta-\epsilon$ proof. Any slip ups in phrasing is purely the instructor's fault. Any punitive action against the student because of an instructor's oversight is a sign of a small-minded instructor.

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I would give this a full grade plus a bonus point. Fantastically elegant solution.

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That depends on the notation used in class. I think your professor wants you to say, 'The limit does not exist' or something similar.

However, in proper context it makes sense to use the notation $$\lim_{x\to\infty}{f(x)}=\infty$$ to mean that the function $f$ diverges. But one should not never say $\infty\cdot\infty=\infty$ in this context, for it implies a sort of extended arithmetic, which is not the usual interest of analysts, but people like set theorists, say.

So, to answer the question. I would mark the student down for saying $\infty\cdot\infty=\infty$ but otherwise note that they understand that the function diverges. That is sufficient. So, a $2.5$?

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    $\begingroup$ I very much hope the professor did NOT want him to say "the limit does not exist", because it exists and is ∞. How can you mark a student down for saying ∞⋅∞=∞? Do you disagree with him? Seriously? $\endgroup$ – gnasher729 Dec 28 '17 at 13:55
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    $\begingroup$ @gnasher729 As I said, that depends on the language, notation & level of rigour they've been using in the course. It's perfectly OK to say a limit doesn't exist in $\mathbb{R}$ if the function diverges. It is abuse of notation, or at best an undefined use of notation, to say the limit is $\infty$. Note that the equation in my answer isn't usually regarded conceptually as an equation. It's only a convenient way of saying the limit doesn't exist. Similarly, it's common practice to use $\lim_{x\to\infty}{f(x)}<\infty$ to mean the limit exists. Again, it all depends on the number space worked in. $\endgroup$ – Allawonder Dec 28 '17 at 15:21
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    $\begingroup$ "However, in proper context it makes sense to use the notation limx→∞f(x)=∞ to mean that the function ff diverges." No, x cos(x) diverges but does not have a limit of infinity. See my comment on Dan Fox's answer. $\endgroup$ – Acccumulation Dec 28 '17 at 22:52
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    $\begingroup$ @gnasher729: This depends on the definition of the range set. Many calculus texts work in real numbers only (not extended reals). E.g., Stein and Barcellos, Calculus and Analytic Geometry, Sec. 2.4: "The notation '$\lim_{x\to 0} \frac{1}{x^2} = \infty$' is useful, though the limit does not exist since $\infty$ is not a number." $\endgroup$ – Daniel R. Collins Dec 28 '17 at 23:48
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    $\begingroup$ I disagree completely. First of all, the arithmetic of the extended real number line is, in fact, very common in analysis. For example, Rudin's Real & Complex Analysis textbook uses this notation throughout. Furthermore, I see no reason that calculus students should be penalized for using this very obvious and intuitive notation. $\endgroup$ – Jim Belk Dec 29 '17 at 16:24
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I'm not a math teacher, but I would give a zero because that's never how I was taught to work with indeterminate forms. What does factoring accomplish, aside from making it look like the student did some work?

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    $\begingroup$ Factoring changed it from infinity - infinity, which could be anything, to infinity times infinity which must be infinite. It is a lovely step. $\endgroup$ – Sue VanHattum Dec 29 '17 at 5:23
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    $\begingroup$ Perhaps you would consider modifying your answer in light of the comment made by Sue above. $\endgroup$ – Amy B Dec 29 '17 at 5:56

protected by quid Jan 2 '18 at 17:54

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