# Question Regarding Limits and Confusing Notation

My calculus teacher assigns us online homework to do. He never went over any question that looks like this (he in fact said we shouldn't be concerned with this):

Yet, I need to answer this right to progress with my homework. It stinks because if I get it wrong, I lose points on my homework average.

So, could someone help explain to me what's going on here, and perhaps guide me to a point where I can try to figure out the solution myself? I'm not asking for a straight answer (although if thats what you want to provide, go for it [since I was told I don't have to know this stuff]), but this stuff really confuses me. Thank you for your help.

Oh, and in case you need it, here's the original prompt (the question I posted above is just a part of a series of questions that go along with this prompt):

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I think the question is quite confusingly worded. It took me several minutes to figure out what it meant -- and it's not as if I don't know the subject matter well.

What must be going on is that you're supposed to imagine reading something like this in a proof:

bla bla bla, and therefore we know that $f$ is continuous, and that $f(c)\ne 0$. We can then apply the definition of continuity with $\varepsilon =$ _______ to find a $\delta$ such that $f(x)$ has the same sign as $f(c)$ for every $x\in(c-\delta,c+\delta)$. Thus, bla bla bla

One of $|f(c)|$ and $|c|$ will make this into a valid argument if you fill it into the blank, and one will produce nonsense. Your task is to select the valid one.

In order to answer the question you need (1) to remember the definition of continuity, and (2) to be able to distinguish a nonsense argument from a valid one. The second of these abilities is often considered too advanced a skill to demand of pre-university students these days (they're supposed to be satisfied with accepting the teacher's judgement in each case), which is probably why your teacher is not allowed to say you must be able to do it...

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Well, another problem I have is that I have no knowledge of what $\delta$ or $\epsilon$ are/means. We were never taught about that matter, and we were even told "not to worry about it." Would you mind elaborating on these terms a little bit? –  Mike Gates Sep 30 '11 at 0:12
Well, then your options seem to be either to read up on it on your own, or to forego the points here. Is the epsilon-delta definition of continuity present in your textbook even if it has not been discussed in class? If so, go read that section. If not, then luckily the symbol usage is fairly well standardized here; Wikipedia's rendering of the definition (under the "Weierstrass definition") should give you what you need. –  Henning Makholm Sep 30 '11 at 0:21
Upon reading this, it seems to me that the solution to my question would then be $\epsilon$ = $|f(c)|$. –  Mike Gates Sep 30 '11 at 0:29
@Mike, correct. –  Henning Makholm Sep 30 '11 at 0:31
Indeed, you are right. But the online stuff is obviously intended for a different type of course than the one you are actually taking. –  André Nicolas Sep 30 '11 at 0:31

Without loss of generality we may assume that $f(c)$ is positive.

Let $\epsilon =f(c)$. By the definition of continuity of $f$ at $c$, there is a $\delta>0$ such that if $|x-c|<\delta$ (and $a\lt c-\delta$, and $c+\delta \lt b$, to make sure we stay in our interval) then $|f(x)-f(c)|<\epsilon$.

Now if you have some experience with inequalities, you should be able to reach the conclusion.

If the "without loss of generality" is not persuasive, if $f(c)<0$, let $g(x)=-f(x)$, apply the above argument to $g(x)$, and see what this says about $f(x)$.

I did not use precisely the language of the question. But you should now be able to see enough of what is going on to be able to answer that question.

Comment: It will be helpful to draw a picture while figuring out what the second paragraph is saying.

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Would you mind elaborating on what the terms $\delta$ and $\epsilon$ even mean? My professor told us "not to worry about those", and the textbook doesn't define them. Some university... –  Mike Gates Sep 30 '11 at 0:16
@Mike: What university, if you don't mind my asking? –  mixedmath Sep 30 '11 at 0:27
If you have not had any prior exposure to these things, it would take a couple of lectures' worth, at least, to get going. The question really cannot make any sense without a fair bit of background. –  André Nicolas Sep 30 '11 at 0:28
@mixedmath: I'd rather not mention it, as not to really trash it's reputation. It's really got a great Computer Science program (and is #1 in its region) [which is my major], but the math department seems lacking. –  Mike Gates Sep 30 '11 at 0:34