a list compiled by Alex Kasman (College of Charleston)
|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:
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:
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
Let's try a more sophisticated example. Consider now
f(2) is undefined
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".
|More information about this work can be found at www.imdb.com.|
|(Note: This is just one work of mathematical fiction from the list. To see the entire list or to see more works of mathematical fiction, return to the Homepage.)|
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(Maintained by Alex Kasman, College of Charleston)