# The Unapologetic Mathematician

## Multivariable Limits

As we’ve seen, when our target is a higher-dimensional real space continuity is the same as continuity in each component. But what about when the source is such a space? It turns out that it’s not quite so simple.

One thing, at least, is unchanged. We can still say that $f:\mathbb{R}^m\rightarrow\mathbb{R}^n$ is continuous at a point $a\in\mathbb{R}^m$ if $\lim\limits_{x\to a}f(x)=f(a)$. That is, if we have a sequence $\left\{a_i\right\}_{i=0}^n$ of points in $\mathbb{R}^m$ (we only need to consider sequences because metric spaces are sequential) that converges to $a$, then the image of this sequence $\left\{f(a_i)\right\}_{i=0}^n$ converges to $f(a)$.

The problem is that limits themselves in higher-dimensional real spaces become a little hairy. In $\mathbb{R}$ there’s really only two directions along which a sequence can converge to a given point. If we have a sequence converging from the right and another sequence converging from the left, that basically is enough to establish what the limit of the function is (and if it has one). In higher-dimensional spaces — even just in $\mathbb{R}^2$ — we have so many possible approaches to any given point that in order to avoid an infinite amount of work we have to use something like the formal definition of limits in terms of metric balls. That is

The function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ has limit $L$ at the point $a$ if for every $\epsilon>0$ there is a $\delta>0$ so that $\delta>\lVert x-a\rVert>0$ implies $\lvert f(x)-L\rvert<\epsilon$.

We just consider the case with target $\mathbb{R}$ since higher-dimensional targets are just like multiple copies of this same definition, just as we saw for continuity.

Now, let’s look at a few examples of limits to get an idea for why it’s not so simple. In each case, we will be considering a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ which is bounded near $\left(0,0\right)$ (since just blowing up to infinity would be too easy to be really pathological) and even with nice limits along certain specified approaches, but which still fail to have a limit at the origin.

First off, let’s consider $\displaystyle f(x,y)=\frac{x^2-y^2}{x^2+y^2}$. If we consider approaching along the $x$-axis with the sequence $a_n=\left(\frac{1}{n},0\right)$ or $a_n=\left(-\frac{1}{n},0\right)$ we find a limit of ${1}$. However, if we approach along the $y$-axis with the sequence $a_n=\left(0,\frac{1}{n}\right)$ or $a_n=\left(0,-\frac{1}{n}\right)$ we instead find a limit of $-1$. Thus no limit exists for the function.

Next let’s try $\displaystyle f(x,y)=\frac{x^4-6x^2y^2+y^4}{x^4+2x^2y^2+y^4}$. Now the approaches along either axis above all give the limit ${1}$, so the limit of the function is ${1}$, right? Wrong! This time if we approach along the diagonal $y=x$ with the sequence $a_n=\left(\frac{1}{n},\frac{1}{n}\right)$ we get the limit $-1$. So we have to consider directions other than the coordinate axes.

What about $\displaystyle f(x,y)=\frac{x^2y}{x^4+y^2}$? Approaching along the coordinate axes we get a limit of ${0}$. Approaching along any diagonal $y=mx$ with the sequence $a_n=\left(\frac{1}{n},\frac{m}{n}\right)$ the calculations are a bit hairier but we still find a limit of ${0}$. So approaching from any direction we get the same limit, making the limit of the function ${0}$, right? Wrong again! Now if we approach along the parabola $y=x^2$ with the sequence $a_n=\left(\frac{1}{n},\frac{1}{n^2}\right)$ we find a limit of $\frac{1}{2}$, and so the limit still doesn’t exist. By this point it should be clear that if straight lines aren’t enough to simplify things then there are just far too many curves to consider, and we need some other method to establish a limit, which is where the metric ball definition comes in.

Now I want to go off on a little bit of a rant here. It’s become fashionable to not teach the metric ball definition — $\epsilon$$\delta$ proofs, as they’re often called — at the first semester calculus level. It’s not even on the Calculus AB exam. I’m not sure when this happened because I was taught them first thing when I took calculus, and it wasn’t that long between then and my first experience teaching calculus. But it’d have to have been sometime in the mid-’90s. Anyway, they don’t even teach it in most college courses anymore. And for the purposes of calculus that’s okay, since as I mentioned above you can easily get away without them when dealing with single-variable functions. They can even survive the analogues of $\epsilon$$\delta$ proofs that come up when dealing with convergent sequences in second-semester calculus.

The problem comes when students get to third semester calculus and multivariable functions. Now, as we’ve just seen, there’s no sure way of establishing a limit. We can in some cases establish the continuity of simple functions (like coordinate projections) and then use limit laws to build up a larger class. But this approach fails for functions superficially similar to the pathological functions listed above, but which do have limits which can be established by an $\epsilon$$\delta$ proof. We can establish that certain limits do not exist by techniques similar to those above, but this requires some ingenuity in choosing two appropriate paths which give different results. There are one or two other methods that work in special cases, but nothing works like an $\epsilon$$\delta$ proof.

But now we can’t teach $\epsilon$$\delta$ proofs to these students! The method is rather more complicated when we’ve got more than one variable to work with, not least because of the more complicated distance formula to work with. What used to happen was that students would have developed some facility with $\epsilon$$\delta$ proofs back in first and second semester calculus, which could then be brought to bear on this new situation. But now they have no background and cannot, in general, absorb both the logical details of challenge-response $\epsilon$$\delta$ proofs and the complications of multiple variables at the same time. And so we show them a few jury-rigged tricks and assure them that within the rest of the course they won’t have to worry about it. I’d almost rather dispense with limits entirely than present this Frankenstein’s monstrosity.

And yet, I see no sign that the tide will ever turn back. The only hope is that the movement to make statistics the capstone high-school course will gain momentum. If we can finally wrest first-semester calculus from the hands of the public school system and put all calculus students at a given college through the same three-semester track, then the more intellectually rigorous institutions might have the integrity to put proper limits back into the hands of their first semester students and not have to worry about incoming freshmen with high AP scores covering for shoddy backgrounds.

September 17, 2009 - Posted by | Point-Set Topology, rants, Topology

1. The only hope is that the movement to make statistics the capstone high-school course will gain momentum.

That movement exists? Thank god! That’s the best (reasonable) thing that could happen to the high school math curriculum right now.

Comment by Qiaochu Yuan | September 17, 2009 | Reply

2. Hi! I’m trying to build an equation editor for the web.

I see that you have a lot of equations directly on your blog. How do you create them?
[Attempt to use my comments section to promote someone else’s business venture snipped]

Comment by Carl Malartre | September 17, 2009 | Reply

3. Carl, like every other mathematician on a WordPress-hosted weblog I type them directly in $\LaTeX$. The WordPress site has clear instructions on how to do this. Your (eventually-)commercial software will not be needed.

Comment by John Armstrong | September 17, 2009 | Reply

4. “The only hope is that the movement to make statistics the capstone high-school course will gain momentum.”

I don’t know how this would serve your purpose of rigor: AP Statistics does not require proofs.

In at least one liberal arts college that I know, first-year calculus (including multivariable calculus) involves no epsilons and deltas: those are relegated to the “Real and Complex Analysis I” course, which most students take after linear algebra (though linear algebra is not really a prerequisite). In this course continuity, limits, etc. are introduced simultaneously for all euclidean spaces (though not for metric spaces, alas).

Comment by Akhil Mathew | September 17, 2009 | Reply

5. Akhil: it’s not about adding rigor to the high school curriculum.

One commonly-advanced argument against including this sort of thing in a given institution’s calculus sequence is that “we’ve got all these kids coming in from high school with all sorts of backgrounds from the AP course and we have to accommodate them all in our higher calculus courses”. Removing calculus of any sort from the high school curriculum puts the entire three-course sequence in one set of pedagogical hands, which is then free to shape the whole thing however it wants. If everyone at Tulane University (for example) has to take the full calculus sequence at Tulane or test out of it by an exam set by Tulane rather than by the ETS — which evidently cannot be trusted to maintain standards — then Tulane is free to include $\epsilon$$\delta$ proofs from the beginning.

Comment by John Armstrong | September 17, 2009 | Reply

• This isn’t really a solution to the problem you’re describing, but $latex\epsilon-\delta$ proofs seem to me like they might be amenable to being treated in a ‘flexible delivery’ (combination of prose and videos) module that people who were interested could do when they felt willing and able.

I think one serious question is how many people who took the old-style courses like you (and me, tho I didn’t actually work through those proofs til much later) ever mastered these proofs (or indeed any kind of proof), or whether they are even teachable at all to the kinds of people who would *not* actively look for and attempt to work through a nice tutorial on the internet.

Comment by Avery Andrews | September 18, 2009 | Reply

6. Well of course that’s the other way to go, Avery: cut out limits and just teach it “Newton-Leibnitz” style.

Comment by John Armstrong | September 18, 2009 | Reply

7. Well I certainly wouldn’t think that to be viable for an whole math program, but where these proofs really ought to go is another matter. Also whether ‘courses’ as such are really the best packaging units, especially for hard stuff such that people can successfully crank through basic exercises but not get the point until rather later (if ever (thinking of myself trying to comprehend monads …))

Comment by Avery Andrews | September 18, 2009 | Reply

8. Sure, not for the whole math program. Just for the calculus sequence. Most programs (around here anyway) have an upper-level “advanced calculus” or “real analysis” course which goes back and covers the same ground, but now proving everything. Get rid of limits in calculus and add them in for the rigorization in advanced calculus.

Comment by John Armstrong | September 18, 2009 | Reply

9. So that would be the kind of program Akhil was talking about, so I guess the question is how well does it actually work?

Comment by Avery Andrews | September 18, 2009 | Reply

• I think what I mean by ‘actually work’ is whether the people who don’t see limits until a intro analysis course actually learn them then, or whether some kind of prior exposure in a Calculus course is helpful to the final result, whatever seems to happen at the time.

Comment by Avery Andrews | September 18, 2009 | Reply

10. Not quite, Avery. Akhil said it involved no $\epsilon$s or $\delta$s. I’m saying to dispense with these messy “limit” things entirely for the calculus sequence. Do it with infinitesimals like dx, the way Newton and Leibniz did.

Physicists and engineers made do without formal limits for literally centuries, and we basically end up telling students that in most cases they don’t need to pay strict attention anyway. So why include it at all if we’re not going to do it right?

But your qualm basically reproduces mine above in a different area: “If we don’t teach [concept A] at this point, will students be able to understand [concept B] at this later point?”

Comment by John Armstrong | September 18, 2009 | Reply

11. Hi John, sorry if it looked like a business venture, but I couldn’t find your email to ask you directly.

By the way, it would be a free service.

I’m still wondering how you create your equations.

Thanks,
Carl

Comment by Carl Malartre | September 18, 2009 | Reply

12. And I already said I use $\LaTeX$. Go to WordPress’ website and search on “latex”.

Comment by John Armstrong | September 18, 2009 | Reply

13. […] doesn’t make a function continuous. Let’s look at the first pathological example of a limit we […]

Pingback by Partial Derivatives « The Unapologetic Mathematician | September 21, 2009 | Reply

14. As a student who took AP Calculus in the 1980’s, I’ll say that although epsilon-delta was taught in into calc, I certainly didn’t really didn’t understand it until I started trying to learn topology, in which case the epsilon becomes an open set in the range, and the delta becomes an open set in the domain, and a limit is simply for every choice of open set epsilon, there exists an open set delta that maps to a subset of epsilon (with suitable restrictions, epsilon has to contain the limit point, etc).

I think that idea, of open sets in the domain and co-domain that map to one another, might be easier to grasp than for every e>0 there’s a d>0 such that for all x, 0<|x-x0|<d implies |f(x)-L| < e.

The epsilon-delta definition limits the open sets to intervals centered on x0 and L, and while that is sufficient (and proofs can be easier with it) it somewhat masks what's really going on, IMHO.

Comment by Buddha Buck | September 21, 2009 | Reply

15. Well, sure there are better and worse ways of explaining it. When I’ve taught $\epsilon$$\delta$ proofs I’ve given a lot of visual intuition, roughly corresponding to the open neighborhoods you’re talking about, and trying to build up why the language of the $\epsilon$$\delta$ approach “means” the “close-enough” heuristic.

And, of course, if you’ll notice in my archives I defined topologies in terms of open sets and neighborhood systems before even touching the real numbers, let alone $\epsilon$$\delta$ proofs or the rest of single-variable calculus back then.

Comment by John Armstrong | September 21, 2009 | Reply

16. “trying to build up *why* the language
means (what it means).”
(my italics.)

right. very important. i like to point out
that here at the beginning we’ll be using
(pictures [graphs] in particular]
to understand the code *in order* that
in future we’ll be able to use the code
in understanding the functions better.

i might first have noticed this in “increasing”
and “decreasing” functions: right in here
concepts that we feel we understand
pretty well already are teased into what
must seem a weird form.
inequalities? quantifiers?

early this year. here, for example.

statistics replacing mathematics is a horror;
a huge win for the enemies of clarity.
could be good for *college* maths at that though.
(fewer clients… less money…)

Comment by kibrolv | September 22, 2009 | Reply

17. statistics replacing mathematics is a horror; a huge win for the enemies of clarity.

I’m not so sure.. First of all, it’s replacing calculus, specifically. And do you really think that AP calculus is a Big Win for clarity? I’ve seen the students coming out the other side, and I’m pretty sure it’s not. Putting the whole calculus sequence in the hands of the same teachers would be.

Comment by John Armstrong | September 22, 2009 | Reply

18. maths is for understanding; statistics is for persuasion.

moreover the statistics party openly abhors proper notation.

a good leaving-alone from the
lies-and-evasion machine
that is public school
would indeed improve
calculus instruction.
no argument there.

AP calc was a “big win”
for nobody but the testing
industry as far as in know.
certainly not for clarity.
still. the powers were pretending
that symbolic reasoning mattered.
they no longer feel that need evidently.

Comment by kibrolv | September 23, 2009 | Reply

19. maths is for understanding; statistics is for persuasion.

This is an interesting aphorism. But if you’ll allow me some aikido, I think it actually makes an argument for statistics as the capstone course. After all, if it does really go “lies, damned lies, statistics” then wouldn’t it be a good idea to teach students how people are trying to lie to them with statistics?

Comment by John Armstrong | September 23, 2009 | Reply

20. […] Now, does the existence of these limits guarantee the continuity of at ? No, not even the existence of all directional derivatives at a point assures us that the function will be continuous at that point. Indeed, we can consider another of our pathological cases […]

Pingback by Directional Derivatives « The Unapologetic Mathematician | September 23, 2009 | Reply

21. what would be a good idea for the students
has so little to do with math ed though.

anyhow, nicely observed.
and again, there’s no argument from me:
in the sense that it messes with
the math maturity of prospective
college students… something like
your original point… so if akido helps it along,
well, that’s show biz!

what’s been *lost*, then?

a major cultural marker
for the scientific revolution.
*newton* as some sort of shakespeare of science
(the english-speaking-people’s greatest hit).
science as the basis
for our age of miracles.
and a nod to the simple fact
that there’s *some* stuff where
opinions count for nothing
(and that this is the *important* stuff).

also:
opportunities for me as a private tutor.

all that. so it’s my own private horrorshow maybe.
no harm intended in inflicting it on you of course.
vive l’unapologeticity.

Comment by kibrolv | September 23, 2009 | Reply

• i *meant* (dammit) to’ve said that
stix messes *less* with math’l
maturity than calc…

some bugs in copy can’t be seen
by their editors until printed.
(everyone who’s ever run off
exams knows this.)

whattya want… an *apology*?

Comment by kibrolv | September 23, 2009 | Reply

22. […] separately. Indeed, now we can even say more about what can go wrong, because these are examples of multivariable limits. We cannot take the limit as and together approach their limiting points along any particular […]

Pingback by Improper Integrals II « The Unapologetic Mathematician | January 15, 2010 | Reply

23. i came across this post because i am having EXACTLY this problem right now as a multivariable calculus student. i don’t think i am incapable of ‘absorbing the logical details’, i am just looking for a good source to learn from… what about using polar coordinates? yet another trick i had to pull off the internet rather than from my prof or textbook…

Comment by jdichter | February 28, 2012 | Reply

24. Polar coordinates are one trick that does work in some situations, but not always. Really there’s no good universal approach but $\epsilon$$\delta$ proofs.

As for you, in particular, being able to pick it up on the fly, you may be right; start practicing on all the limit exercises in your textbook. But I (and many other multivariate calculus instructors) have tried and it’s just not a viable approach for most students who have no experience from a previous single-variable calculus class.

Comment by John Armstrong | February 28, 2012 | Reply

25. As a statistician, I shudder at the thought of making statistics a capstone high school course. I had a year of graduate econometrics, and I still didn’t really begin to understand statistics until I worked my way through a calculus-based text. As with limits, the problem is that key statistical concepts are much easier to understand with formal development (maximum likelihood is a prime example). But, even in my highly-rated grad program, most students had not taken calculus.