|An 8 minute long, black and white film with no dialogue showing intertwined scenes of a student having trouble with the concept of a limit in his calculus class and other scenes from his life. The director has sent me the following summary:
A mathematics student, Gabriel Lontane, has a hard time understanding what a “limit of a function” is. While mentally re-examining his student life memories, he sets off on a slow walk through the university campus: a metaphor of life and our vain attempt to get close to things. This experimental short film attempts to sow within its narrative a number of visually descriptive examples of what the mathematical concept of the limit of a function purports. The story is based on few excerpts from the book “Voiceless Lilies” written by the director.
I'm afraid that the film may serve to further disseminate a common misconception about the term "limit". If one thinks that the word "limit" in mathematics means something about "not being able to reach something", then the film contains some arguably artistic analogous scenes in the student's non-mathematical life. For instance, we see a moving shot of a small girl reaching through a fence to get a ball that is just out of reach, and we see Gabriel's girlfriend trying to reach him on the phone when he is out on his walk.
However, this is not what the word limit means in mathematics. Cristiano is not alone in his misunderstanding. Not only many students, but many calculus teachers seem to have trouble with this. And so I generally make a concerted effort to make it clear when I teach calculus. There are, I think, two reasons that people who learn just a bit about limits think of it in terms of being "unable to reach" something:
- In standard English, the word "limit" does have a meaning which implies a boundary beyond which one cannot pass. This, however, should not be confused with the different meaning that the same word has in mathematics that does not imply anything about an impassable boundary.
- The technical definition of limit is not terribly abstract, but it is one of the more abstract mathematical terms that a non-mathematics major will encounter. It says this: "I will say that the limit as x goes to a of f(x) is L if for any positive number c that you pick, I can find another number d such that the distance between f(x) and L is less than c for every number x (other than a) whose distance to a is less than d." The parenthetical remark "other than a" seems relatively unimportant to me, but it seems to be the focus of many students (and less sophisticated teachers) of mathematics. Believe me, the rest of it is the important part and as you will note, it does not contain any "limitations" in the English sense. The technical definition of limit basically says that "the limit as x goes to a of f(x) is the number that f(x) gets close to as its input gets close to a." We merely exclude the value of f at a itself.
On the contrary, I would say that the wonderful and amazing thing about the "limit" in mathematics is that it allows us to reach things we otherwise could not! Let me show you all of this with a few examples. (I'm assuming here that the reader already knows something about functions and limits, but I will try to explain it clearly enough for someone who does not.)
A completely trivial example is to consider the function f(x)=x2-4 and ask "what is the limit of f as x goes to 2?" That means, what happens to the values of f as the values of x approach the number 2. We can get an idea of what it would be by plugging in some values near 2 and looking for a pattern:
f(2)=?we don't look at this
Notice that I did not evaluate the function at x=2. This is the only significance of the "but not equal to a" comment that seems to become the entire focus in the misunderstanding of limit. But, so what? Look at what pattern we get from the other numbers. Coming down from above, we see that as the input numbers approach 2 the output numbers get smaller and smaller. In fact, since we get more zeroes at the front of f(x) when the number of zeros between the 2 and the 1 in x increases, we expect that f(2.000000001) would be a number that starts .0000000... and hence would be very close to zero. This is in agreement with what we see when we go up from the bottom. The number 1.999 is closer to 2 than 1.99 is, and that is closer than 1.9. But notice again that the closer the input is to 2 the closer the output is to zero. In fact, it is true that the limit of this function as x goes to 2 is zero. Does that mean that we are somehow unable to reach zero? No, we can also compute f(2) and see that it is 22-4 which is in fact equal to zero.
Let's try a more sophisticated example. Consider now
This function is a little less pretty. Note, for instance, that it is undefined at x=2 because f(2) would have a denominator of 2-2=0 and you cannot divide by zero. Notice that this limitation of not being able to plug in x=2 is NOT a "limit" in the mathematical sense. It is just a problem in the definition of the function itself. The limit will, in fact, allow us to get around this problem! Using my knowledge of limits (specifically, using derivatives which are a particular limit and L'Hospital's rule which is a general rule about taking limits) I can determine that
the limit as x goes to 2 of f(x) is in fact equal to 4.
We can check this by plugging in some values of x near to 2:
f(2) is undefined
Yes, it does seem that if f(2) was equal to 4 then it would fill in this hole in the table.
In fact, that is probably the best way to think of a limit in mathematics: it is a way to fill in a hole. If there is no hole (as in the case of the trivial example with f(x)=x2-4) then the limit doesn't do much for you. But, when there is a hole in the graph of a function (in the second case, the graph was a nice smooth line except at x=2 where it had a little hole) the limit allows you to fill it in (if you add a point at (2,4) on the previously described graph it fills in the hole and connects the two parts of the graph so that they now form a single smooth curve)!
The important role that it plays in differential calculus is exactly that. We want to find a formula for the slope of the tangent line to a graph (which we will call the derivative). However, the formula for slopes that we know which involves only algebra (which would be [f(x)-f(a)]/[x-a]) has a hole at x=a, which is exactly the one we want. Algebraically there is nothing that can be done about this, but with the limit, we can fill in the hole and thereby define the derivative!
Anyway, I'm not sure if my description of the limit would make a good movie, but I think it is a much more optimistic view of the mathematics itself. Remember, even if "limit" sounds like a depressing word in English, in calculus it is more like a superpower. It allows you to find values you need which ordinary mortals (using only algebra) could not obtain! This is the basis for calculus, and hence for many of humanity's greatest achievements, from the use of radio waves for communication to space travel.
The director of the film is a professor of language at the University of Toronto and the script appears in his book "The Graviton, The Millinery Man".
I just read your article on the Infinitely Near page you've created and wish to
thank you for such a detailed and helpful explanation.
I also wish to reiterate that Infinitely Near is the story of a student who has
difficulty in understanding the very “points” or “holes” mathematics classes'
focus on when it comes to the limit of functions. In the film the one who has
no such difficulty is the character of the professor (played by Patrick
O'Donnell, who happens to be a real physics professor at the University of
Toronto). I've preferred to use the case of a polygon made to approximate the
circumference of a circle as the model function to ‘explain what a limit is.'
The professor speaks to his class about it at the beginning of the film. Later
he meets with Gabriel, and makes the coin spin (seen in a close-up shot later
on in the film) to further illustrate the points he lectured on, which are
represented with other functions notes appearing on the blackboards.
The original script of the film (which appears in The Graviton book) does
contain some lines of dialogue (though that is not a necessary condition to
call it a ‘script'). Film scripts may contain only descriptions of scenes
without dialogues. In the process of shooting the film I've modified the story
line from the original script. In the editing I found the silent experiment
(actions and music score) visually and psychologically more interesting to my
story. The idea of ‘getting close to things' is the psychoanalytical struggle,
if you will, the student goes through to understand his math lectures and what
is happening to him in his life experiences with his girlfriend.