The epsilon delta proof allows us to prove that a 'given' limit is actually the limit. In the sense that, we can show using that if a limit for a sequence/ function exist, then that limit is a unique one. But, how does one find a limit following the epsilon delta definition or do we need something more?
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asked Jul 24, 2022 at 7:08
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2$\begingroup$ FYI: Eventually, you'll learn about Cauchy sequences, where what matters is that terms get arbitrarily-close to each other (roughly speaking ) rather than arbitrarily-close to a predetermined value $L$. In appropriate contexts, this is enough to ensure convergence to a limit without having to know the limiting value ahead of time. $\endgroup$– BlueJul 24, 2022 at 8:48
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$\begingroup$ Mmm I have. But that is only in complete metric space. Good point. $\endgroup$– tryst with freedomJul 24, 2022 at 8:50
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$\begingroup$ "complete metric space" = "appropriate context". :) $\endgroup$– BlueJul 24, 2022 at 8:52
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2$\begingroup$ Your question seems to boil down to "How do you compute limits?" (as "finding the $L$" in an $\varepsilon$-$\delta$ proof is precisely the task of computing the limit). This is an incredibly broad question. I don't see how it can be reasonably answered---even the accepted answer her only gives very vague advice. $\endgroup$– Xander Henderson ♦Jul 24, 2022 at 15:38
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1$\begingroup$ There are many techniques for computing limits, and many theorems about limits. Generally, we use those theorems and techniques, and are taught to use them in analysis classes. But there is not (nor can there be) a technique which will allow you to compute every possible limit. $\endgroup$– Xander Henderson ♦Jul 24, 2022 at 19:45
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1 Answer
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It depends on the limit. Sometimes, it's obvious and you can just eyeball it. Sometimes, you have to multiply and divide by the relevant quantity in order for it to become obvious. Sometimes, you have to just look at the graph to get a sense of what the limit is.
Sometimes, you actually may have to apply formulas or theorems which are not necessarily applicable just to have an answer to work with. Keep in mind that theorems have conditions and if the conditions are not satisfied by whatever it is that you're dealing with, then you cannot use that theorem. You either have to do something else or you have to modify the thing you're dealing with sufficiently to apply the theorem.
But there's no law that tells you that you can't just sneakily cheat and apply a theorem where it may not be applicable just to get a sense of what the limit is. There's no law that's telling you that you can't just approximate functions by their Taylor expansions and just use that to get a rough estimate for what the limit is.
All of this can be done on scrap paper and, so to speak, is hidden away from public view. We don't publicly tell everyone that we found a limit by L'Hopital's rule. We just tell them what the limit is and prove that it is the limit via the golden standards of proof-writing, whether it be by an $\epsilon-\delta$ argument or by making use of other theorems (by the way, making use of theorems to prove a limit where it is justified is perfectly rigorous.)
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$\begingroup$ With a good number of math problems, there is an element of guessing going on. The goal is to gain enough experience so that the guessing game becomes a lot more organized and the eventual goal is to become as systematic as you can be. $\endgroup$ Jul 24, 2022 at 7:48
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$\begingroup$ Some teachers will present a neat, polished proof and give the students no idea at how it was obtained. I try to avoid this and allow my students to see my guesses and false starts. I just have to hope that I get there before I run out of time. $\endgroup$– badjohnJul 24, 2022 at 9:14
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$\begingroup$ @MordeusMorgenstern: Although many students may do it wrongly, using asymptotic expansion can be done completely rigorously. Just look at the relevant links from my profile, which also explain roughly how to do it systematically. Properly programmed CASes like Maple use asymptotic expansions to compute limits as well. $\endgroup$ Aug 25, 2022 at 10:06