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0 pointsconcreteblock5y ago0 comments

As someone who has taught intro calculus a few times, one reason for the emphasis on computational techniques is simply that 90% of the students in such a class do not care/are not capable of grasping the epsilon-delta definition of the limit.

Solution: spend more time on epsilon-delta so that students have time to wrap their minds around the idea. But I think the engineering departments would complain that the students who we send on to them cannot do basic computations. Also students would complain that we spend too much time on theory and not enough on application. There are probably other reasons that someone more experienced would know about.

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javajosh5y ago· 2 in thread

I tutored calc too, but I had an unanswered complaint about the epsilon/delta definition of a limit - I found it to be circular. There was no impact on the computations required by the class, but the definition seemed rather circular, as it presupposes the reality of the infinity implicit in induction. Over time I recognized that my intuition was rebelling against unphysical math concepts. I was a physics major and so I was content to hand-wave my own concerns away as a merely philosophical problem, since you could always pick a sufficiently large number of induction iterations to satisfy any practical need. But yeah, it left a bad taste in my mouth that no-one seemed to care about this issue. (Oddly the use of induction for similar things, like Cantor's diagonal argument, didn't bother me at all. It was specifically epsilon delta that did, and still does, bother me.)

rsj_hn5y ago

The epsilon delta definition of limit is not circular and there is no infinity anywhere in it. Are you referring to the limit as something goes to +infinity on the real line? That's just bookeeping shorthand for some quantifiers over bounded numbers.

I'm genuinely curious as to what you were getting hung up on, because odds are you learned the wrong definitions or somehow changed the definition to something that is incorrect, and then discovered the error later on. It's a fairly straightforward definition if you can do the proper bookkeeping over the quantifiers, and is designed to not have anything to do with infinities. But many people struggle with quantifiers.

a-dub5y ago

this is the point. they do not stress in beginning calculus that continuity is actually a profound property of a mathematical construction (the reals) that we use for modeling physical phenomena.

it's not the delta-epsilon definition that is slightly weird, it's what it means to be truly continuous that is. (uncountable infinity, or as i like to call it, endless zooming everywhere)

once that is clear, the delta-epsilon definition becomes obvious and clear.

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