Grading a limit problem

Question

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.

Answer 1

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.

Answer 2

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?

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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.

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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.

Answer 11

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.

Answer 12

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.

Answer 13

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.

Answer 14

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

Answer 15

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 $2.5$?

Answer 16

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?