I have always wondered whether it is necessary or not. For me, it seems that it is enough to teach them the intuitive idea, that is, limit is just an approximation of a certain process.
what do you think about it? I am basically interested in knowing if there is pedagogical reasons based on the nature of their training.
I hope you get my concern because my english is not very good looking.
Well it doesn't really feel right to get degrees in engineering and gain years of engineering experience without even knowing what a limit actually is. And even though many engineers will do just fine without having been exposed to the rigorous definition of a limit, some engineers will need to be familiar with rigorous definitions/proofs if they ever pursue a career in academia or if they ever have to read textbooks/papers that haven't been specifically written for engineers.
It is probably not the place of mathematics educators to decide what mathematics courses engineering majors should take. But a good reference point is ABET accreditation. Over 600 universities in the US have ABET accredited engineering programs. We should defer to the professionals who set these standard and assess outcomes. Here is a description of their current criteria:
They refer to college level mathematics, and say "For illustrative purposes, some examples of college-level mathematics include calculus, differential equations, probability, statistics, linear algebra, and discrete mathematics." Accredited undergraduate programs require a minimum of 30 semester credit hours of college level mathematics and basic sciences. This document gives more specific expectations for different types of engineering programs. For example, Civil Engineering requires mathematics through differential equations.
I think from this it is clear that a "survey course" in calculus would be inappropriate for engineering students. Students in engineering programs are expected to take the same sort of single variable calculus course taken by other STEM majors, including math majors.
The concepts behind limits are actually very important to engineering (in the form of error/precision analysis), but are rarely phrased that way.
Given a function $f$, we can imagine an engineering situation where there is some desired range of outputs from the function, but the engineer has control over the value of the inputs of the function. If $f(c) = L$, one might want to set the value of $x$ close to $c$ so that $f(x)$ is within some error $\epsilon$ of $L$. But since this is a real-world engineering situation, one cannot set the value of $x$ precisely, only aim for a target value and specify a level of precision. That is, we should aim to make $x = c$ and hopefully there is some level of precision $\delta$ we can set on $x$ so that if $x$ is within $\delta$ of $c$, then $f(x)$ is within the desired range.
One can then ask about the relationship between the precision necessary on the input to produce a desired precision on the output. For differentiable functions, this relationship is linear (in the limit to 0), and the multiplier is the derivative.
No, it's definitely not "necessary".
I'm not an engineering major, but roomed with one, did a general engineering minor, and worked in/around mechanical, nuclear, mining and chemical engineering (had electrical on staff too). Passed my EIT and was at one time, about to take the PE (mechanical) exam. Most engineers in the workplace don't even use calculus, let alone epsilon-delta. For that matter I've worked with a (small number of, but still violates a Euclidean "necessary") licensed PEs who didn't even have a college degree.
In their normal undergrad training (fluids, thermo, etc.), engineers will often use calculus. So you need calculus to get through a BS in E. But you don't need or use epsilon-delta in those courses, either in the derivations or the homework drill or the projects.
Even if we use a less picky interpretation of "necessary" (very helpful), I would not put epsilon delta up there. For one thing, the current stereotypical STEM training in the US, gives an exposure to epsilon delta, with a few problems, at the beginning of a calc course. But doesn't come back to it, doesn't use it. So it's not even that important for undergrad calculus or diffyQs.
Now, does it hurt to have something like this? I don't think so. Especially given the limited time spent on in a typical calculus course based on a text like Thomas. I mean at least you've seen it so someone mentions it, it's not a mythical creature. And it's really just a bunch of detailed algebra pushing symbols and equations around (my same feeling about series)--maybe builds the algebra muscles in a manner helpful for and similar to multistep homework problems in fluid hydraulics or power systems. But it's not firmly connected--just general muscle building. For the vainshingly small but not converging to zero population of undergrad engineers that go on to some theoretical Ph.D. (and even within that a small set of them) and turn out to need this sort of stuff, they at least were exposed to it. And can pick it up more as needed, for their work. (Not to say they couldn't even with no previous exposure.)
Being practical, we can always come up with things that might in some circumstance be helpful (learning Latin, say). But life is finite. It is practical. It's an engineering problem, with constraints, costs. etc. So I definitely would not use the strong word necessary (even if we use it colloquially to mean strongly important) to describe formal limits. There are a gazillion things useful to engineers and a lot on their slate already. There is limited time in the day and limited brains in the skull. So, I wouldn't go beyond the week or two at beginning of a calc course, as done now.
