What's the purpose of Epsilon proofs for limits?

[Deveno]

i think it's more a matter of information organization (boy, THAT's a mouthful).

for example, in linear algebra, it's more "empowering" to really understand what a basis is good for, than to have lots of computational experience calculating eigenvalues and the like. it's not that computation isn't useful, or that working with n-vectors, matrices (and eventually tensors) is "wrong" it's that the "practical application" parts of linear algebra can obscure the ways in which the full power of linear algebra can be used (apparently, for example, actuaries use it all the time).

a lot of people learning calculus for the physical sciences (or even, gasp! biology), will still encounter, even if at a somewhat less-intense level, functions of more than one variable. if limits for single variables seem intractible at times, multivariate limits can get downright funky. and epsilon-delta views of things really do help wade through the fog.

there's a certain sort of give-and-take that occurs in math: if the definitions are hard, the theorems are easy, and vice versa (note: there are counter-examples). i think the topological concepts (and the geometric intutions you can use along with them) are given short shrift when limits are "glossed over".

but sure...for an average calculus student, it's straight from the quadratic formula and a little trig to...wtf? open set? least upper bound? where did my nice algebra that i was finally getting the hang of go? what's an "existential quantifier" and why is it bad if i get the "for all" and the "there exists" in the wrong order?

the thing is: a little density can go a long way. topological methods are starting to show up in the most unlikely of places: statistical analysis (like demographic analysis for advertisers), security system design, biological classification. a metric is a powerful concept, and it's useful for more than just "number-crunching" types of math.

perhaps in the class the thread starter is taking, the limits being examined are ones that aren't too pathological, reinforcing the idea that "usually" we can do without all this complicated stuff. personally, i like the idea of basing calculus on "nearness" (which actually implies the epsilon-delta definition) and is closer in spirit to the general requirement in topological spaces that f^-1(U) is open if U is. "neighborhood" is a nice, friendly word, and helps loosen up the formality of the language.

khan academy has some really nice videos where he breaks down visually where the epsilon and the delta comes from. when you see it graphically, it makes more sense. i'll admit, there's a certain dryness and lack of humor permeating calculus, all of a sudden it's: stuff just got real.

*****

f(x) = 1 - (x²)^1/3 was what i meant. silly me.

 

[homeomorphic]

for example, in linear algebra, it's more "empowering" to really understand what a basis is good for, than to have lots of computational experience calculating eigenvalues and the like. it's not that computation isn't useful, or that working with n-vectors, matrices (and eventually tensors) is "wrong" it's that the "practical application" parts of linear algebra can obscure the ways in which the full power of linear algebra can be used (apparently, for example, actuaries use it all the time).

I'm all for a more theoretical approach, in general, with the caveat that it's an ideal and the students are not ideal. We're arguing over a very specific point, here. I see it as quite out of place to come to more general conclusions about a general philosophy of teaching, here. I'm advocating a more intuitive approach, which means mindless computation is de-emphasized.

a lot of people learning calculus for the physical sciences (or even, gasp! biology), will still encounter, even if at a somewhat less-intense level, functions of more than one variable. if limits for single variables seem intractible at times, multivariate limits can get downright funky. and epsilon-delta views of things really do help wade through the fog.

Well, the tricky limits for multivariable functions that I am aware of are the ones I saw in my analysis class in undergrad. You may have a point here, but you'd have to investigate whether this really comes up in applications or not and whether it's really necessary to use delta epsilon or is it enough to have counter-examples and intuition. From what I've seen of physics and engineering, I never had to take any weird multivariable limits, and being a student often involves more theory. But my experience is limited.

The only completely reliable way is to be rigorous, but rigor isn't some kind of magic thing that automatically prevents all mistakes, either. The only way in which it is completely reliable is if you don't make any mistakes (when you just work intuitively, it may not always be 100% clear if you have made mistakes or not). So, it's slightly circular to argue that it's going to prevent mistakes, especially give that most non-mathematicians find it very difficult. So, part of the problem is that, normally, it's a moot point whether you teach epsilons and deltas or not because most of the students don't get it, and even if they do, they may soon forget.

there's a certain sort of give-and-take that occurs in math: if the definitions are hard, the theorems are easy, and vice versa (note: there are counter-examples). i think the topological concepts (and the geometric intutions you can use along with them) are given short shrift when limits are "glossed over".

That's a different issue. It may be correlated with but is not caused by glossing over the precise definition is of a limit. As I said, we're talking about something very specific, namely, deltas and epsilons--that is maybe one section of the textbook and 1 or 2 lectures.

but sure...for an average calculus student, it's straight from the quadratic formula and a little trig to...wtf? open set? least upper bound? where did my nice algebra that i was finally getting the hang of go? what's an "existential quantifier" and why is it bad if i get the "for all" and the "there exists" in the wrong order?

Yes, if you're going to do it, do it right. Don't slam them with things they are unprepared for. If the students are miserable and don't learn it anyway, all you will accomplish is to build resentment.

the thing is: a little density can go a long way. topological methods are starting to show up in the most unlikely of places: statistical analysis (like demographic analysis for advertisers), security system design, biological classification. a metric is a powerful concept, and it's useful for more than just "number-crunching" types of math.

It takes time to learn all that. There are different kinds of engineers. Some engineers like math. If they want to get a double major, that is great. I recommend it. I am not discouraging anyone from pursuing math to whatever degree they wish. If they are really mathematical, then they can pursue an academic career in control theory and be just like a mathematician. All I'm saying is if they aren't interested in it, it's optional whether they want do epsilons and deltas. That's not true quite for everything. They have to do certain things in order to be good at what they do and they may not like 100% of what they have to do. They have to decide whether the bad parts of the career outweigh the good, and if so, they can do something else.

khan academy has some really nice videos where he breaks down visually where the epsilon and the delta comes from. when you see it graphically, it makes more sense. i'll admit, there's a certain dryness and lack of humor permeating calculus, all of a sudden it's: stuff just got real.

That's a good point. Even though I actually have a very intuitive understanding of the epsilon-delta definition, perhaps, in my manner of speaking, I have set up a false dichotomy between that stuff and intuition (though I was well aware of this all along). Epsilons and deltas have their own intuition.

f(x) = 1 - (x2)1/3 was what i meant. silly me.

Yeah, not differentiable at every point. You can see it visually if you plot it. And the plot is understandable. You don't have to take the computer's word for it.