Wikipedia:Reference desk/Archives/Mathematics/2008 June 5

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June 5

Limits

I am studying the basics of limits and I have two questions:

1.In calculating limits of some functions, you will have the situation that the limit is undefined since it has to be divided by 0. For example:

$\lim _{x\to \ 3}{\frac {\sqrt {x-3}}{x^{2}-9}}$

You can't get the limit unless you eliminate the x-3 somehow. But basically, what changes is not the value of the fucntion but what it looks like. I feel like I have skipped the problem of zero division rather than solve it right through the limit. The feeling I have is similar to the history that people couldn't notate some numbers because they didn't have the concept of zero. Is there also some imperfection in modern notating system so that we don't know how to manage zero division?

2.limit of a function should be "a value that a function can infinitely APPROACH to". This implies that the function is not necessarily continuous at the limit. However, the process of limit calculation is just substitution, as if the function can surely reach the limit without being undefined. Isn't there any problem?

--Lowerlowerhk (talk) 05:03, 5 June 2008 (UTC)

I think your two questions are very much related, and I hope that what I say below will help you. It is not true that the process of limit calculation is just substitution. Let's consider some function f(x) and consider the limit as x tends to zero. Then think of f as undefined at zero (even if it is) and figure out the value to assign to f at zero so that it would become continuous. This would be the limit then. A very simple example would be ${\frac {x-3}{x-3}}$. It is undefined at 3, but the only way to define it at 3 in order for it to be continuous is to set it to 1. Oded (talk) 05:42, 5 June 2008 (UTC)

(edit conflict) The best way to think of it is that the value of a limit is equal to the number you approach when you choose a value at a point arbitrarily close to the intended point, without actually choosing that point. That is, you pick a point close to the given point, then you pick one that's closer, and one that's closer than that, and so on. The smaller the distance is, the closer the number is to the actual value of the limit, regardless of what would happen if you actually tried to evaluate the function at the target value. As for your question on "imperfection in modern notating system", see the article on infinitesimals. « Aaron Rotenberg « Talk « 05:50, 5 June 2008 (UTC)

In cases like this you can use L'Hôpital's rule, I will not go over the specifics unless you wish me to, but in a nutshell you can use it where substitution would give ${\frac {0}{0}}$ or ${\frac {\infty }{\infty }}$. Rambo's Revenge (talk) 12:25, 5 June 2008 (UTC)

Woooah ... in the case of

${\frac {f(x)}{g(x)}}={\frac {\sqrt {x-3}}{x^{2}-9}}$

L'Hôpital's rule does not help us because

$\lim _{x\to \ 3}{\frac {f'(x)}{g'(x)}}$

does not exist, and so this tells us nothing about the original limit (L'Hôpital's rule is an if, not an if and only if). Anyway, there is a simpler algebraic solution - cancel the common factor from numerator and denominator to get a function that is identical to the original function except at the point x=3, then use the behaviour of the new function as x approaches 3 to draw a conclusion about the existence of the original limit.

Returning to Lowerlowerhk's original questions, I suspect that some of the underlying confusion may be due to their exposure to an informal and non-rigorous treatment of limits and continuity. Throw away the "infinities" and work through a proper epsilon and delta treatement of limits, and all will become much clearer. Gandalf61 (talk) 12:57, 5 June 2008 (UTC)

Of course if we do that, i.e. cancel the common factor, you get a function that by inspection you can see doesn't converge as x->3.Richard B (talk) 13:48, 5 June 2008 (UTC)

Getting back to L'Hôpital's rule for a second, doesn't it still apply if $\lim _{x\to \ 3}{\frac {f'(x)}{g'(x)}}$ is $\infty$? Specifically, if $\lim _{x\to \ 3}{\frac {f'(x)}{g'(x)}}=\infty$ then doesn't it follow that $\lim _{x\to \ 3}{\frac {f(x)}{g(x)}}=\infty$? 63.95.36.13 (talk) 16:03, 5 June 2008 (UTC)

The rule actually does apply here: note that the requirement is only that the resulting "derivative limit" take on a value in the extended reals, so an infinite result does carry back to the original limit. --Tardis (talk) 16:05, 5 June 2008 (UTC)

Yes, I stand corrected. But I still think that l'Hôpital's rule is an unnecessarily baroque approach for this particular problem. Gandalf61 (talk) 16:27, 5 June 2008 (UTC)

Woooah, I opened a can of worms here. Whilst theoretically it can be applied, i did overlook the precise question, and practically it is not the best method. Apologies Rambo's Revenge (talk) 17:14, 5 June 2008 (UTC)

We might want to talk about left and right limits. Note that the original expression is imaginary for x<3, so there is no (real) left limit. For the right limit:

$\lim _{x\to 3^{+}}{\frac {\sqrt {x-3}}{x^{2}-9}}=\lim _{x\to 3^{+}}{\frac {\sqrt {x-3}}{(x+3)(x-3)}}=\lim _{x\to 3^{+}}{\frac {1}{(x+3){\sqrt {x-3}}}}=+\infty$

--Prestidigitator (talk) 18:37, 5 June 2008 (UTC)

And, since positive infinity isn't a real number, there's no real right limit, either... --Tango (talk) 19:50, 5 June 2008 (UTC)

Mmm...You can't say the value of a function is infinite somewhere, but an infinite limit is fine as far as I know. --Prestidigitator (talk) 20:13, 6 June 2008 (UTC)

There's nothing wrong with functions that have infinite limits, or for that matter functions that take infinite values. But in neither case is the answer a real number, which was Tango's point. It's an extended real number. --Trovatore (talk) 20:22, 6 June 2008 (UTC)

This was placed on the science desk..you guys might be able to answer it

What is the name of this equation and what is it's significance? What does it mean? $\langle \Omega |{\mathcal {T}}\{{\hat {\phi }}(x_{1})\cdots {\hat {\phi }}(x_{n})\}|\Omega \rangle ={\frac {\int {\mathcal {D}}\phi \phi (x_{1})\cdots \phi (x_{n})e^{i\int d^{4}x\left({1 \over 2}\partial ^{\mu }\phi \partial _{\mu }\phi -{m^{2} \over 2}\phi ^{2}-{\lambda \over 4!}\phi ^{4}\right)}}{\int {\mathcal {D}}\phi e^{i\int d^{4}x\left({1 \over 2}\partial ^{\mu }\phi \partial _{\mu }\phi -{m^{2} \over 2}\phi ^{2}-{\lambda \over 4!}\phi ^{4}\right)}}}$ Ζρς ι'β' ^¡hábleme! 05:15, 5 June 2008 (UTC)

I think it means, "I really wanted to show off when I was writing this equation". Or maybe, "Nah, nah, my brain is bigger than yours!" « Aaron Rotenberg « Talk « 06:23, 5 June 2008 (UTC)

Holy shit, is that an integral inside an exponent inside an integral???? Someguy1221 (talk) 06:44, 5 June 2008 (UTC)

Wisdom89 _{(T / ^C)} 06:52, 5 June 2008 (UTC)

I think this actually belongs in the physics desk. This looks more like something a physicist would write. More precisely, it looks like statistical physics / quantum statistical physics / conformal field theory. I would not assume that this is necessarily show off. Oded (talk) 09:49, 5 June 2008 (UTC)

Yeah, looks like physics to me. I think some context would be useful - where did you find the equation? --Tango (talk) 14:32, 5 June 2008 (UTC)

I found it on Uncyclopedia and YouTube; however, it has been verified as a legitimate equation on the Science refdesk. Note: I didn't cross-post this so you don't have to tell me not to. I don't know who did though. Wisdom 89 did. Ζρς ι'β' ^¡hábleme! 19:23, 5 June 2008 (UTC)

I think I've seen this before in a book about prime numbers. I think it's something to do with Srinivasa Ramanujan; although I'm most probably wrong... Jonny23415552 (talk) 19:52, 5 June 2008 (UTC)

Guessing: The left hand side is a Dirac bracket of quantum mechanics. Ω is a state vector and T is an operator. The right hand side includes covariant tensor derivations of a potential Φ. The denominator is a normalization constant that makes the fraction a probability. The integration inside the exponent is over a four-dimensional volume. The m is a mass and the λ is the cosmological constant. The value of this integral is a phase. The Planck constant does not enter explicitely into the equation, so the units chosen make the Planck constant equal to unity. So this may be an equation attempting to express quantum gravity. I am merely guessing. Bo Jacoby (talk) 21:53, 5 June 2008 (UTC).

Someguy1221: single-variable ODEs tend to produce integrals in the exponential when you solve them. But why on earth does this formula have two bars in the angle brackets? – b_jonas 07:49, 9 June 2008 (UTC)

See Bra-ket notation#Linear operators. —Ilmari Karonen (talk) 14:22, 9 June 2008 (UTC)

Ah, I see. Thanks. – b_jonas 19:07, 10 June 2008 (UTC)

Looks fake to me for a number of reasons, but mostly because the integrals do not appear to integrate with respect to anything, for example $\int x\cdot dx$ unless if there is some type of maths I haven't seen before where this is correct. Adamd1008 (talk) 20:19, 10 June 2008 (UTC)

Just for a sense of completeness of the archives I'll post this 6 months late. It is indeed a physics equation from the realm of Quantum Field Theory. The integrals are 'functional integrals' (sums of a functional over all functions \phi belonging to some set) which is why it doesn't look like a standard integral. They formally denote this by $\int {\mathcal {D}}\phi$.

In fact the equation given is an example of an operator expectation value being calculated in a "massive scalar \phi^4 theory" which is often used for didactic purposes. See Scalar field theory for a bit more information.

Explicitly, they are doing something similar to this for the action, S[\phi], given by this and a functional, F[\phi] given by the time ordered field operators T{\phi ...}.

Hope that cleared it up a little. Peter Ballett (talk) 01:14, 21 December 2008 (UTC)

Operator

What does this operator mean (the "\bigoplus" one)? $V=\bigoplus _{n=0}^{\infty }V_{n}$ --Ζρς ι'β' ^¡hábleme! 19:23, 5 June 2008 (UTC)

Direct sum. --Tango (talk) 19:49, 5 June 2008 (UTC)

(ec) The "\bigoplus" is just a big version of "\oplus"; it can be used to depict iterated application of the operator, like the sigma is used for summation. Your formula $\bigoplus _{n=0}^{\infty }V_{n}$ is thus equivalent to $V_{0}\oplus V_{1}\oplus V_{2}\oplus \cdots$, as you probably know.

The interpretation of $\oplus$ itself depends on context; I've seen it used for the XOR operation and for the intersection of constraints in constraint satisfaction literature (the latter is thus identical to the intersection operator $\cap$). Oliphaunt (talk) 19:56, 5 June 2008 (UTC)

With the Vs, though, it's extremely likely to be the direct sum of vector spaces. Algebraist 23:03, 5 June 2008 (UTC)

See also Wikipedia:Reference desk archive/Mathematics/2006 August 18#What is this called?.  --Lambiam 05:14, 7 June 2008 (UTC)