# Do you think that answering simple limit questions with taylor expansions or asymptotic notation should be downvoted or frowned upon?

I've seen a lot of questions of rather elementary limits (most of them 'highschool' level), where users answer saying things like

"Note that $\sin x =\sum \frac {(-1)^n}{(2n+1)!}x^{2n+1}$, then it's rather trivial $\dots$"

Or

"Just remember that $\sqrt x \in o(x)$, then $\dots$"

I'm always disappointed at seeing answers like that, it's utterly obvious they'll be useless for the one asking the question (and most of the times it's really kind of "killing an ant with a nuclear bomb").

What do you guys think?

• Why is it obvious that such answers will be useless for the one asking the question? – Daniel Fischer Mar 30 '16 at 1:11
• I don't consider asymptotic notation to be particularly advanced math... Sure it may not be well-known in high school, but it just means $\lim_{x \to \infty} \sqrt{x}/x = 0$, this isn't exactly rocket science if you know what limits are. – Najib Idrissi Mar 30 '16 at 7:40
• Really, this is why we ask users to provide context to their questions: so that we can give an answer of the right level to help them. If they don't do so, it's practically impossible to tell what they know about. – Chappers Mar 31 '16 at 3:20
• On the rather narrow issue of Questions that ask for limits "without l'Hôpital", I object to Answers using Taylor series as an evasion that amounts to essentially a circular argument. – hardmath Apr 3 '16 at 14:55
• If the question is being asked by someone with limited math skills, then there is nothing wrong with using Taylor series and this may well have been intended for the approach to be taken. – jim Apr 4 '16 at 20:30
• I am somewhat surprised that nobody mentioned this older discussion yet - it seems to be related, at least to some extent: Using Taylor expansion on a limit tagged “without L'Hospital” – Martin Sleziak Apr 8 '16 at 4:25

I think such answers should not be downvoted merely because they are beyond the presumed understanding of the OP. (There may of course be other reasons to downvote the answer.) In the first place, "answers are for everyone." I can't count the number of times that I've learned something here because someone else introduced a more powerful technique than I was aware of. (Well, obviously it's countable, but—oh, you know what I mean.) In the second place, who can say for sure what the OP knows, or can apprehend once shown?

As for frowning upon such answers—to be sure, no one can know if you frown upon something, but in my opinion, it's not productive to censure someone (not censor) for using a sledgehammer instead of a flyswatter. My countersuggestions:

• Let the OP comment on the answer if they don't find it helpful; their feedback is more likely to elicit a positive response from the answerer than someone from the peanut gallery.

• Upvote the answers you do approve of, rather than downvote the one you don't.

• If no one else has provided a more elementary solution, it may behoove you to add one.

1. You have no way of knowing if the answer will truly be useless to the one asking the question.

2. "killing an ant with a nuclear bomb" So what? Is it a bad thing?

3. Answers are for everyone, not just the OP. Maybe these answers will be helpful to someone else. Considering that questions get closed for being duplicate of one another, if someone asks a question about some limit but displays high-school level knowledge, does it mean that everyone is forbidden from ever answering this limit using power series on this site?

• 1) You can't know, but you can often make a reasonable guess. (2) It doesn't help people who want to kill ants but don't possess nuclear bombs. (3) Simpler accessible answers are more useful (for everyone), not just for the OP – bubba Apr 2 '16 at 5:42
• @bubba 1) I thought math was about certainties? 2) Again, what about the people who do want to use a nuclear bomb? You're being egoistical... If you don't want to read answers that use advanced concepts, don't read them then, instead of forbidding everyone from writing one. 3) What? I spent years of my life learning complicated math, why should I now forget it? The underlying math concepts may be simpler, the resulting proof may not be. A two-line conceptual argument is simpler and more accessible than a three paragraphs "elementary" proof. – Najib Idrissi Apr 2 '16 at 5:58

Well, I certainly frown on these sorts of answers, personally. If nothing else, they show a lack of care and sensitivity towards the OP. I sometimes leave a comment telling the author that his (or her) answer is probably useless to the OP. And in extreme cases, I downvote.

I think it's often possible to judge the OP's level of mathematical ability and sophistication from the content and phrasing of the question. Providing an answer that's clearly way over the OP's head is just plain rude (to the OP), in my view.

I also think the "answers are for everyone" argument is a weak excuse for overly abstract or sophisticated answers. The more knowledge you assume, the less accessible the answer. Not just to the OP, but to everyone.

• Did you somehow forget that a question can receive multiple answers? What harm if done to OP is someone answers the questions with high school level tools, and someone else answers using more advanced ones? None at all. If an answer is really completely inaccessible, then nobody will yogurt it. Why would you frown upon a question because "well, I understand it, but most people won't so it's useless"? That's just condescending. – Najib Idrissi Apr 2 '16 at 6:08
• How autocorrect changed "upvote" to "yogurt", I'll never know. I'm leaving the comment as is for posterity. – Najib Idrissi Apr 2 '16 at 9:40
• @NajibIdrissi: As someone with only a vague, high-level understanding of mathematics (and no, I'm not in school any more, so I'm not going around asking homework questions), I strongly appreciate this stance. If I can understand the question, I should be able to understand the answers in most cases (but not all; see Fermat's Last Theorem and other such things). – Kevin Apr 3 '16 at 4:57
• While "care and sensitivity towards the OP" is laudable, once a sufficiently elementary answer has met that need, even the OP (and more importantly the future Readers of any really good Question) are apt to benefit from answers that use more general or powerful arguments. – hardmath Apr 3 '16 at 14:49
• @NajibIdrissi -- You really ought to consider becoming a lawyer. You have quite a knack for distorting other people's words. I didn't say that an overly sophisticated answer is useless,I said it was inconsiderate to the OP. I said that the more knowledge the answer assumes, the less people it will help. That seems obvious. What did I say that is condescending? – bubba Apr 30 '16 at 8:25

In general I won't down vote an answer just because the answer is an overkill. That kind of answers might provide the asker a new way of thinking the problem that (s)he never sees before (like asymptotic equivalent in your examples).

I'll, however, consider down-voting an answer if the answer use a technique that the question explicitly stated to not use (e.g. Use L'hopital when it's stated in the question that it's not allowed).

Contrary to what you think, the community seems to dislike the opposite- trying to crack a hard problem using elementary approach while using more advanced tool can yield a simpler and more natural solution. For example, the following answers I wrote that got down-vote(s).

example 1

example 2

example 3

• Another example: math.stackexchange.com/questions/1261036/… My answer was the first and uses essentially only elementary combinatorics, but it's has the lowest score, whereas the sledgehammer answer has the same score as the accepted answer. – Matt Samuel Apr 12 '16 at 11:04

It is not necessary that answers are useful to the author. Nice side effect when it happens, but it is not a reasonable expectation to impose on the question/answer dialogue.

Fortunately, almost any correct answer that is not an extreme digression from the point could be useful. If a simple limit question gets answers using power series, that can be a hint to anyone interested to learn about power series, and a valuable connection to remember whenever the learning about power series happens. "These things can be used to compute limits! Where did I see that before...?".

• the answer should be useful to the person asking the question otherwise what is the point of asking the question? "Everyone" includes the author. – Q the Platypus Apr 5 '16 at 3:20
• The point of a Stackexchange question is to invite viewers of the page to compose an archived searchable resource on that topic. That this sometimes is useful to the OP is a side effect whose only relevance to math.stackexchange is that the OP (or others who like the "useful to OP" side of things) might add upvotes or a green check-mark on a "useful" answer. That's all. There is nothing wrong with answers that are pertinent as a resource on the proposed topic but happen, for whatever reason, to not satisfy the author of the question. – zyx Apr 5 '16 at 14:35
• Wow. Amazing. No wonder I sometimes feel alienated around here. It never even occurred to me that value to the OP is merely a "side-effect" of some grand scheme to accumulate mathematical knowledge. I thought we were supposed to be providing the OP with an answer to his (or her) question. I thought that was the point of all this. – bubba Apr 9 '16 at 13:30
• Glad to be of help. There is nothing in the MSE software to stop people who think of themselves as servants to anyone who types a question to a web site, from acting in accordance with that belief. As long as the rest of us are equally free to post whatever form of (correct, archived, searchable, topical) resource we wish to under a question, without complaints and meta threads saying that this is "not helping the OP enough", there is no problem. The second use mode is not any less helpful to OPs than the first, and is cleaner in that there is no concept of obligations or favors. @bubba – zyx Apr 9 '16 at 18:36
• I disagree strongly that helping the OP is an optional "side effect" of an answer. Ideally, the answer is targeted to the OP. – Cheerful Parsnip Apr 10 '16 at 0:28
• @GrumpyParsnip: I don't think that zyx would disagree that the answer is ideally targeted to the OP, but would disagree that it is a requirement on the answer, especially if an answer suitable for the OP has already been posted. At any rate, discouraging such answers seems to me to lose more than it gains: Even if there isn't already a more suitably elementary answer, posting a somewhat complex answer may elicit comments from the OP that will more precisely illuminate what concepts they are or aren't familiar with. – Brian Tung Apr 11 '16 at 17:19
• There's nothing wrong with an answer being targeted to the OP, and nothing wrong with answers that are targeted only at some aspect of the question (without considering what OP might ideally want). The second type of answer can and often does help the OP as a side effect, and does not stop anyone from trying to directly help the OP, but I think the Stackexchange software is really not built to be a tutoring platform, and having specific OP-help on a large scale or expected for every question would start to become a problem. Splitting off a tutoring site could solve that. @BrianTung – zyx Apr 12 '16 at 22:58
• @zyx: I quite agree. – Brian Tung Apr 13 '16 at 1:50