User talk:Rschwieb/Sandbox

Comments on the adapted proof

Shouldn't m(x) be the infimum (g.l.b.) and M(x) the supremum (l.u.b.)? The interval should be an open finite or infinite interval with c as an extremity; c may be finite, +∞, -∞. Also, you shouldn't be using the endpoint c except in the limit:

${\frac {f(x)-f(c)}{g(x)-g(c)}}={\frac {f'(\xi _{x})}{g'(\xi _{x})}}$

use x & y in I, instead. Toolnut (talk) 19:17, 13 December 2011 (UTC)

Typos fixed. Rschwieb (talk) 20:24, 13 December 2011 (UTC)

The interval ${\mathcal {I}}$ has to be open, at least as far as c is concerned, for you to be able to say that $g'(x)\neq 0{\text{ on }}{\mathcal {I}}$. Recall in your counterexamples to me that $g'(x)$ could equal 0 at x=∞, while L'H's rule still applies for $\lim _{x\to \infty }g(x)=\infty$. The same problem occurs when you select x & y from ${\mathcal {I}}$, where the endpoint c should not be a choice. Taylor makes that clear at the beginning of his proof: "Let ${\mathcal {I}}$ be an open interval ..." Toolnut (talk) 17:55, 14 December 2011 (UTC)

I cannot find any valid point in your post. The article currently says "The real valued functions f and g are assumed to be differentiable on an open interval with endpoint c" and my proof says "Let the interval be as in the hypotheses." I do not have to "able to say" that g' is nonzero on the interval, that is a hypothesis. At no point is c a choice for anything in the proof. One step I can take though is to emphasize the openness of the interval again in the proof. My counterexample with "derivative 0 at infinity" does not apply, becase Taylor and I are making no attempt to use the extended real numbers, as you wanted to.

I appreciate your help, but I do not appreciate this embarassing retaliatory tone you are no longer even trying to hide. I hope you are not taking it personally that homemade user proofs are forbidden by WP policy. I'm also sorry if I didn't sound very friendly when pointing out problems in your own proof: I was just pointing out things that didn't make sense as written, without knowing how the real proof went. This process has taught me quite a bit about the details, and has been a rewarding experience for me, so I owe you for drawing me in. Let's call it square and move on. Rschwieb (talk) 03:28, 15 December 2011 (UTC)

Were you talking about the c in the footnote? That was an accidental overloading of c. I swapped it for a p. I use c for the intermediate point in the mean value theorem in my lectures. Rschwieb (talk) 03:35, 15 December 2011 (UTC)

No, I wasn't. I'm sorry, I did not see the hypotheses you speak of in the main article, the last time I looked: it looks like you added that to it a day ago. Still, it's better that you've emphasized that point in the proof.

Please don't get angry at me: I don't know what I've done to deserve it. I've always thanked you for your participation. This is the first time you've shown any appreciation for what I've been doing: it's time-consuming thankless enough work for someone currently unemployed like me, with little hope of employment doing something I love doing, such as this. Toolnut (talk) 04:42, 15 December 2011 (UTC)

There is no problem here. I would be happy to begin looking at your work in progress again, if you decide to invite me back. Even though it won't be mainspace content, refining it will go a long way to increasing your proofwriting skills. Also, thanks for catching the infinity symbol problem. I have this big moniter, but no matter how big I blow up the text, all the math symbols below look TINY! I can hardly tell them apart... Rschwieb (talk) 19:26, 15 December 2011 (UTC)

You are invited, always have been. I considerably trimmed down the proofs when I exploited the fact that lim f(x)=∞ is superfluous, as it has just become known to the both of us. Otherwise, the proofs remain the same and I'll put them down as nothing more than a good learning experience. No need to waste time analyzing them further, at least I don't have that luxury, anymore: I'll need to get back to earning a living, one way or another, sometime soon; my math tutoring is only bringing in about $80/mo. right now. I am nearing retirement age with hardly any savings to speak of. Toolnut (talk) 02:26, 16 December 2011 (UTC)

You did invite me, but then I sort of uninvited myself. Since the situation was resolved and with your approval, I'll be glad to help. Good luck with your job search! Rschwieb (talk) 14:55, 16 December 2011 (UTC)

More Questions

Couple of points:

1. Maybe you'd want to mention that "$g(x)\neq g(y)$ for distinct x & y follows from $g'(x)\neq 0$" (as Taylor has done), or that $g'(x)\neq 0{\text{ on }}{\mathcal {I}}$ is an aditional condition in Cauchy's MVT, in

   From the differentiability of f and g on ${\mathcal {I}}$, Cauchy's mean value theorem ensures that for any two distinct points x and y in ${\mathcal {I}}$ there exists a ξ between x and y such that

   ${\frac {f(x)-f(y)}{g(x)-g(y)}}={\frac {f'(\xi )}{g'(\xi )}}$

2. Since you went through the trouble of establishing that we can obtain the condition that $g(x)\neq 0$ in the open interval by sufficiently shrinking that interval, based on the given $g'(x)\neq 0$, something Taylor doesn't see a need to do, why not go a step further and show that the latter given condition follows (or not?) from the given that $\lim _{x\to c}{\frac {f'(x)}{g'(x)}}$ exists.

Hint: If $g'(x)=0\,$ somewhere in the interval, however much we shrink it, then $g'(x)\,$ must be oscillatory at c about 0. Unless $f'(x)\,$ also oscillates and crosses zero at the same time as $g'(x)\,$, ${\frac {f'(x)}{g'(x)}}$ would be oscillating b/w -∞ and +∞ ad infinitum and this would contradict the given existence of that ratio's limit. However, if they both oscillate and their zero crossings coincide, it gets complicated. I'll let you ponder that one over, if you wish.

Toolnut (talk) 16:43, 15 December 2011 (UTC)

• Re1: I went ahead and put in the note you suggested.
• Re2: Taylor did not need to trouble himself about showing that because he assumed it as a hypothesis. Since it is provable in a footnote, I decided to keep the hypotheses to the simple and standard set of "f and g differentiable, g' nonzero near c, and lim f'/g' exists". It will not be possible to remove the g' nonzero hypothesis. Every text and scholarly paper I've picked up so far includes it, and if it were superfluous someone would have definitely said something by now. Before this project, I didn't realize it was necessary at all! Shame on me for not reading these undergraduate math calc texts more closely... Rschwieb (talk) 19:38, 15 December 2011 (UTC)

One last thing

I believe Taylor's paper had a typo that you tried to explain away:

As y approaches c, both ${\frac {f(x)}{g(y)}}$ and ${\frac {g(x)}{g(y)}}$ become zero, and therefore

$m(x)\leq \liminf _{x\rightarrow c}{\frac {f(y)}{g(y)}}\leq \limsup _{x\rightarrow c}{\frac {f(y)}{g(y)}}\leq M(x)$

(For readers skeptical about the x 's under the limit superior and limit inferior, remember that y is always between x and c, and so as x approaches c, so will y. The limsup and liminf are necessary: we may not yet write "lim" since the existence of that limit has not yet been established.)

The inequality should be corrected to

$m(x)\leq \liminf _{y\rightarrow c}{\frac {f(y)}{g(y)}}\leq \limsup _{y\rightarrow c}{\frac {f(y)}{g(y)}}\leq M(x),$

(and the explanation changed) as there is no reason why we'd want to make x→c in liminf and limsup at the same time as y, or before the arguments of the functions m and M, shown to also be x. The only reason why ${\frac {f(x)}{g(y)}}$ and ${\frac {g(x)}{g(y)}}$ both →0 is because y→c before x does, and to a greater degree, so that g(y) is, at all times, much greater than g(x) & f(x), otherwise those two would not necessarily vanish. The limit superior and inferior are not functions of x, but limits, possibly distinct (in case the ratio is oscillatory as c is approached, i.e. the limit does not exist), and therefore should be independent of x, just as the author must have intended when he said "We then let x→c ..." (empahasis added). This typo had me scratching my head for a while, leading me to misunderstand what limit superior and inferior meant: are they functions of x, because x is still the argument of the functions m and M? When it finally occurred to me, with further reading, that those limits are not functions of the variable under the limit sign (similar to the integration variable of a definite integral), I had by then forgotten the cause of my initial misunderstanding; unitl now. Note what would happen if we'd let x & y → c together, while allowing their difference to vanish at the same time, say as (c-x)/2:

${\frac {{\frac {f((c+x)/2)}{g((c+x)/2)}}-{\frac {f(x)}{g((c+x)/2)}}}{1-{\frac {g(x)}{g((c+x)/2)}}}};$

would the limit as x→c be the same? At least, it is not clear unless we stress the fact that y→c, first, independently of x, and then we let x get as close as necessary to c without actually reaching it. Toolnut (talk) 21:19, 17 December 2011 (UTC)

When I saw how it was written, I too wondered why they chose x's over y's. It could truly be a typo. Since you have made the argument for y's too now, I'm inclined to change them. What's in Taylor's paper probably isn't wrong per se, but I do agree it seems clearer to use y first. It will still make sense, and eliminates the need for me to explain away why they are not y in the first place. Rschwieb (talk) 21:47, 17 December 2011 (UTC)

If anyone is still in doubt, here's a simple example that fails to reach the desired limit, when y, which at all times remains nearer to c than x and is therefore distinct from x, thus meeting the only requirement set for it in the proof, is a function of x:

$g(x)={\frac {1}{x-c}},\ y={\frac {x+c}{2}}\neq x,\ \lim _{x\to c}{\frac {g(x)}{g(y)}}=\lim _{x\to c}{\frac {(x-c)/2}{x-c}}={\frac {1}{2}}\neq 0$

For the case where c=∞, all we need do, to show that g(x)/g(y) does not vanish, is let x-y be finite, as x→∞. It's possible that this erratum was later caught by peer reviewers (if not by the author) and published as corrigenda in a later issue. Toolnut (talk) 10:01, 18 December 2011 (UTC)

The corrections you made, up to this point, still hang on to y after stating that

$m(x)\leq \liminf _{y\rightarrow c}{\frac {f(y)}{g(y)}}\leq \limsup _{y\rightarrow c}{\frac {f(y)}{g(y)}}\leq M(x),$

which ought to eliminate y from further consideration (other than to treat it as a generic limit variable that may later be substituted for x to match the desired final form given in the theorem): the only consideration remaining from that point forward is that x be as near to c as necessary to "squeeze" the liminf and limsup into becoming one with the common limit of m & M ("common" because lim( f ' /g ' ) is given to "exist"); "squeeze" is a misnomer here, because the liminf and limsup are already determined and are not functions of x; all the squeezing does is it shows that the two limits, reached independently of x, have to be the same as the common limit of m & M. Bottom line, I would delete the following part:

When x is allowed to approach c in the next step, y will also approach c because y is between x and c. For this reason, the occurences of y will be replaced with x.

Toolnut (talk) 11:36, 18 December 2011 (UTC)

Yes, it's certainly important that y and x be allowed to limit independently. You've convinced me that less is better, so I took out the sentence. It's probably not so much of a typo in the original paper as a quirk of Taylor's writing style. I couldn't find any trace of an erratum on it, and really a journal would probably not waste space correcting it.

Nothing is wrong with the use of the Squeeze theorem, although admittedly this is not its normal use. There is no rule against the squeezed function being constant or already having a known limit. Ostensibly it is the logical conclusion of an argument bounding a quantity between two functions known to share a limit at a point. Did you think of an alternative way to force equality between the three numbers? Rschwieb (talk) 21:07, 18 December 2011 (UTC)

No, you're right that there is no restriction to the squeeze theorem, that the "squezeed" function not be constant, as in this case (perhaps, I was rash in calling it a misnomer). But I believe using analysis (the "ε" method) would make this proof more robust, and would avoid trying to explain, in words, why y has to be "infinitely closer" to c than x for the undesirable terms to vanish uniformly (independently of x) from Cauchy's equation in the "∞/∞" case; the way we've avoided doing that was by making y go to c first and noting that what remained was no longer a function of x. In case you're interested, I have an improved step-by-step analysis method in my latest Proof 2. Toolnut (talk) 12:54, 19 December 2011 (UTC)

Avoiding epsilons would certainly be preferable in this article. Proofs are only rarely admissible, and absolutely rigorous proofs are even less desirable. I'm pretty happy with the changes we've developed so far. It has all the main finesses of Taylor's proof, plus some good crosslinks to relevant WP articles. Rschwieb (talk) 17:58, 19 December 2011 (UTC)

Update: Rereading your comments above finally crystalized the nagging doubts I have been having about the y's in the liminf and limsup. I think that Taylor's notation is correct. You are under the impression that the liminf and limsup are independent of x, but as I thought before, the fact location of y beteween x and c and x 's approach to c are significant to mention. I think Taylor means $\liminf _{x\rightarrow c}{\frac {f(y)}{g(y)}}=\sup(\inf\{{\frac {f(y)}{g(y)}}:y{\text{ is between }}x{\text{ and }}c\})$. This isn't necessarily a constant sequence, but rather a function of x that is nondecreasing as x approaches c. Similarly, using inf(sup(-)), the limsup is a function of x. It does not conform exactly with the WP definition of liminf, but I know you are no stranger to modified notation. Rschwieb (talk) 19:29, 19 December 2011 (UTC)

You're trying at all cost to justify an obvious typo in Taylor's proof: why do you think your interpretation of liminf and limsup, which does not at all conform to WP's or the worldwide community's, should be obvious to any reader? Why not insert the explanation you just made to me into the article and see how much approval it shall receive there? I may be "no stranger to modified notation" but at least my notation is not as counterintuitive as this one is; but what about the ordinary Joe reading the article? It violates established rules and is counter-pedagogic.

A limit, any limit that exists, is a constant w.r.t. its variable, which is taken to the limit and which I call the "limit variable", in this case clearly stated to be x (as in x→c): x should not feature in the result once the limit is taken. Therefore, the limit cannot be a function of the limit variable, x. Toolnut (talk) 07:56, 20 December 2011 (UTC)

• I'll try to express what's in the article in notation closer to the WP article, but I am nearing my limit of willingness to modify the original source. I hope you haven't forgotten the limits imposed on editors of WP articles already.
• As a person of borderline competence in the subject, you should give Taylor and the professional reviewers a little benefit of the doubt. Rschwieb (talk) 14:37, 20 December 2011 (UTC)