I've mentioned several times that one of my school's calculus teachers and I are taking a specialized calculus class at a nearby community college. This class generated a spirited texting debate between the two of us.
I was watching yesterday's class on limits, and the instructor was doing some example problems about what happens to functions as x approaches plus or minus infinity. There were times when he'd say that the limit was "infinity", with which I disagree. If a function's end behavior is such that the function goes off to plus or minus infinity, then the limit does not exist. I base my position on the definition of a limit, which includes an epsilon/delta proof.
My friend and colleague agreed with our professor, saying that stating a "limit" is infinity describes the end behavior of the function. While I agree, it's not technically a limit; if a function grows without bound then it's not limited.
I broke out my Swokowski textbook from my calc class in 1982, which supported my position (although not explicitly), and he broke out a later Swokowski and then a Larson and a Stewart textbook, all 3 of which stated explicitly that a limit could be infinity. I'm convinced Swokowski was right the first time and later sold out!
As a math major I was trained formally. A limit requires a real (and finite) value L, as well as an epsilon/delta proof. I understand that saying "the limit is infinity" is a convenient shorthand, and for those of us who know what we're doing, it's useful but not formal. For beginning calculus students, we should teach the rules before we teach how to break them--sort of like we do in grammar.
Not to be outdone, I went to my Advanced Calculus book, from the semester course wherein we proved every single thing we took for granted back in elementary calculus. There was nothing about "infinite limits" in that text.
We kinda sorta agreed that it may not be formal but it was practical. Two of my closing texts:
Utility vs. formality. I can oscillate like a sinusoidal between those two poles.
and
Formally, the limit does not exist, because the function approaches infinity...But as I said, I'll survive saying the limit is infinity.What do you say, math geek readers? Does a limit equal infinity, or not?
(And yes, math teachers really do have such discussions.)
11 comments:
I think you nailed it when you said, "convenient shorthand". A limit has to approach the same CONSTANT NUMBER from both sides. Infinity is not a number...it is a concept.
When I read this, I had a vague memory of an analysis class where we worked with an extended set of the reals- we added + and - infinity to the reals. Then it made sense to say that the limit was infinity or the limit at infinity was 4, for instance. As I recall, in this extended set, you could use something like epsilon-delta proofs to show infinite limits.
Looking it up today, I find this construction is called the affinely extended real numbers. Being too lazy to look for a textbook reference, I'll just point you to Wikipedia for an overview. Apparently, it is possible to generalize the Cauchy sequence construction of real numbers to include +/- infinity. The resulting construction is no longer a field (because, for instance, infinity has no inverse), and +/- infinity are not real numbers.
My analysis professor, as I recall, was of the opinion that such an extension of the reals was, to borrow an old mathematical expression, "as refined as it [was] useless." Which is likely why my memory was so vague- she was an excellent teacher, and I remember most of the class much more vividly.
I believe what I said to Darren was that when we say that the limit of a function as x approaches a number from right/left/both is infinity, we are implicitly stating that the limit does not exist and, at the same time, describing the behavior of the function as x approaches that value, which is extremely useful (think vertical asymptotes).
As a person who does complex variables, I like to think of the real line as embedded in the Riemann sphere (this creates the one-point compactificaton of the real numbers). So in this case, we "define" the extended real line by adding infinity, and a limit that, in the calculus sense, diverges to infinity can be said to have a limit _equal to_ infinity (the new point we added).
This also allows one to think of the graph of y=1/x as continuous, by defining 1/0=infinity and 1/infinity=0. Then as we approach zero from the left, the graph goes to infinity (downward along the large negative real numbers), and "wraps around" the point at infinity to reappear at the large positive real numbers. I think I first encountered a formal proof of this in Buck's Advanced Calculus, but it might have been in Ahlfors' complex analysis instead.
Shorthand.
I agree in a sense: You can't have a limit (end) of infinity because it's not an end per se.
But what's the alternative? When you are discussing limits then either you need to invent a new term for "limits" which are infinity, or you need to distinguish between "undefined as a limit because infinity" and plain old "undefined".
Those add much more confusion, IMO, than the supposedly-confusing question of using "infinity" to define the limit of y=1/x, as x approaches 0.
Especially since all you have to do is literally explain this ONCE to the class: "We use infinity as a shorthand. But infinity isn't a number and can't technically be a limit, it's just a lot easier to deal with if we say that a limit is infinity."
First, I would agree with Darren on the semantics that the term limit infers an actual end point, and there is no end to "infinity". Because infinite implies going on forever. This reminds me of a discussion I have had over the years with Math and Science friends. So, as such Mr. RotLC, does the value of 0.9999[infinitely repeating] equal 1?
Does it *equal* one? Or is it a *limit* that equals 1? I don't know the formal answer but I lean towards the latter.
"Approaches infinity" even bothers me, somehow, though I can't say why - and I have probably even used those words! I think I'd prefer "increases without bound" or "becomes arbitrarily large".
"So, as such Mr. RotLC, does the value of 0.9999[infinitely repeating] equal 1?"
Yes, insofar as it is mathematically indistinguishable from 1 and could in theory be used as a substitute for 1 in any equation, without altering the end result.
Anonymous, then why the decimal point, signaling the value is less than one? I claim it approaches one but never actually gets there. And the statement "mathematically indistinguishable from 1 and could in theory be used as a substitute for 1 in any equation, without altering the end result" is not true for y = 1 / (1-x). I can substitute x with 0.999*, but I cannot use 1.
"[T]he statement 'mathematically indistinguishable from 1 and could in theory be used as a substitute for 1 in any equation, without altering the end result' is not true for y = 1 / (1-x). I can substitute x with 0.999*, but I cannot use 1."
What do you propose the result to be if you put in 0.999... into 1/(1-x)? For that matter, what do you propose is the value of (1-x) if x=0.999... How does it differ from 0?
From one perspective (the one I learned), when we write a decimal expansion for a number, we are using a shorthand way of writing a sequence, with the number represented by the decimal expansion being the limit of that sequence. So the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, 0.9999, 0.99999, 0.999999 ... which can be shown by traditional epsilon-N methods to be 1.
Another way of looking at it: if 0.999... is not 1, there must be a real number between it and 1. In fact, since the rationals are dense in the real numbers, there must be a rational number between it and 1. But there is no way to write such a number (which could be done if such a rational number existed).
Post a Comment