Talk:Calculus/Limits/An Introduction to Limits

Requests
While it's asked how | lim [x -> 0]{sin(x)/x} = 1 | can be proven, this isn't demonstrated - there also isn't an actual demonstration elsewhere online, nearly as I can determine. Shouldn't one be put in? --68.212.77.222 23:05, 26 Apr 2005 (UTC)

Is there anyone interested in writing How to evaluate the limit of a real-valued funtion ? --wshun

I think we should put the formal definition of the limit right at the end of page. As it is the most difficult concept to full grasp. And we need some excercises.

The last example seems to imply that f(x) = (x^2 - 9)/(x - 3) is not the same as g(x) = x + 3. Why is this true?

Because you can't divide by zero; g(3) = 3 + 3 = 6, whereas f(3) = (3^2-9)/(3-3) = (0/0) = undefined. The functions will, however, be equal at all other points. (yes I edited your question to make my answer easier to articulate. :) )

BTW, I would consider completely deleting the introduction. It makes some questionable statements, and I can't see it as being anything but confusion for someone who doesn't know anything about limits already. --Xhad

Am I the only one that thinks that this whole thing is worthless without the Cauchy definition of a limit? You cannot proove any of the statements unless you define limits properly. The best way to define them is the Cauchy definition!!

arbitrary break
(x^2-9)/(x-3) has the problem of a loss of information as x approaches 3. The solution to this is rearranging the problem so that you are no longer dealing with having a small number on the denominator. This is the realm of computational mathematics, though, and I'm not sure if it's useful to put in a Calculus textbook. --Ronald Raygun

To me that sounds like something that would be appropriate for a Calc primer for physicists, but it seems beyond the scope of a typical Calculus course. --Xhad

It would be neat if someone would implement a parser that would feed this latex math notation to a graphing program... easy visualisation on the click of a function. -- Kowey 09:03, 4 Oct 2004 (UTC)

Can any-one tell me that if we are taking x->0 and f(x)->L//then why are puting there an = in the answer when it should be ~(approximate value) --221.135.232.166 10:19, 27 June 2007

To answer the above question, it is because of the definition of a limit. Although we think of $$ \lim_{x \to c} f(x)$$ intuitively by the process of taking values of $$x$$ closer and closer to $$a$$, the limit is not formally related to the value of $$f(x)$$ at $$x=a$$, which may not even be defined. For instance, consider the function defined as follows:

$$ f(x) =\left\{\begin{matrix} 0 & \mbox{if } x\neq 1 \\ \mbox{27} & \mbox{if } x=1\end{matrix}\right. $$

By the definition of limit, $$ \lim_{x \to 1} f(x)=0$$, which is a really bad approximation of $$f(x)$$ at $$x=1$$. Greenbreen (discuss • contribs) 13:15, 16 March 2011 (UTC)

Possible inclusions for this page
- limits at infinity (horizontal asymptotes) - Infinite limits (e.g. vertical asymptotes) - Left/right handed limits - thinking of limits of indeterminant forms  (those that can be evaluated without Lhopitals rule)  - proofs of limit rules/one side limit theorem

Maybe this page needs to be broken up. Continuity is a whole topic, and the formal definition/proofs of limit rules could also be pulled to its own page. -- julius

dutch
Hey there, I'm making a dutch version of this book/chapter. I think I may have solved some presentational problems so if you like check out http://nl.wikibooks.org/wiki/Analyse:Limieten. Robert Zboray

LEDE
The section named Finding Limits starts as follows: ''Now we will concentrate on finding limits, rather than proving them. In the proofs above, we started off with the value of the limit. How did we find it to even begin our proofs?'' I find this confusing. There are no proofs above that I can see, only examples. And in those examples we did actually find the value of the limit. (I realize that this error is probably a rest from a previous re-dispositional edit..) Or am I missing something? In that case I apologize. /Hugo --83.181.53.162 20:31, 5 November 2006 (UTC)

You CAN divide by zero
EVERYONE LISTEN TO THIS. I'M TIRED THAT EVERYONE KEEPS TELLING THAT YOU CAN'T DIVIDE BY ZERO. Of course you can; it's a limit.

lim x/y = a. When y approaches zero, a = infinity.


 * Technically, you can divide by y in the limit as y approaches zero. You cannot divide by zero without the use of such a limit. Remember that:
 * $$\lim_{y \to 0} y \ne 0$$
 * The limit never allows y to actually take the value of zero, only to approach that value by infinitessimal amounts. As such, y never equals zero, but instead equals a value infinitessimally close to zero. The two quantities, while close, are not strictly the same. --Whiteknight (talk) (projects) 20:33, 9 January 2007 (UTC)

Whoa! I'm going to have to set this straight. You can't divide by zero, neither algebraically nor analytically. In algebra, the defining quality of the additive identity, zero, is that it has no multiplicative inverse (unless the set is trivially {0}).

PROOF:
 * Assume that zero has an multiplicative inverse, and let 0 * a = 1.
 * b*a + 1 = b*a + 0*a = (b + 0)*a = b*a  (assumption, distributive prop., def. of additive identity)
 * which means that 1 = 0, which is a contradiction whenever the set is not {0}.

Let's look at it analytically now. Remember, a limit has to be well-defined to exist. If it takes on different values depending on the way you approach the limit, then the limit doesn't exist. Back to your example:
 * lim(y->0) x/y. Without loss of generality, let x > 0 (if x = 0, then the limit is 0, and if x < 0, we can multiply it by -1)
 * lim(y->0 from the left-negative side) x/y < 0 and lim(y->0 from the right-positive side) x/y > 0
 * Therefore the limit is not well-defined. It doesn't exist.

Now, you could try to patch it by definition, but no matter what you'll be left with some "infinity" which isn't accounted for. Also, negative infinity is not equal to positive infinity... that's easy enough to see.

Lastly, to the point: $$\lim_{y \to 0} y = 0$$. If one uses the formal definition this is obvious. There is no actual "approaching" - that's just a simple way to think about it. "The two quantities, while close, are not strictly the same" - They are only different in that they have different names. (Sketch: Let the infinitesimal x (>0) be a number different from 0, so that x/2 is not 0. Then |y - 0| < x/2. Obviously y cannot be x, for x <= |x| = |x - 0| < x/2 is a contradiction.) The infinitesimal is a nice tool for investigating functions of limits, but technically it's not a number in the sense you're trying to use it. Mo Anabre 06:43, 1 May 2007 (UTC)


 * I did simplify too much in my explanation, i should have taken the time to explain it properly. --Whiteknight (talk) 17:37, 1 May 2007 (UTC)

Hello! Intuitively, zero is a special symbol (like $$\infty$$), that denotes empty expression. But if there is abscence of anything, what are you trying to compare with divident? Of course, division by zero can be defined artificialy, but AFAIK all such definitions lead to contradictions with other axioms. Exanode 15:16, 7 November 2007 (UTC)

There are two limit of each function (though sometimes both give the same result), one from the positive side and one from the negative side, so actually you can take the limit of x/y as y approaches 0. This is also a mistake in the text, which says that the limit of 1/(x-2) as x approaches 2 is undefined. The limit of this function from the negative side is negative infinity and the limit from the positive side is positive infinity.

TO: authors.
Please put some visualizations in the text. It's best to describe things like limits with graphics and give viewers an intuitive visual analogy.

I've seen some very interesting eLearning math courses with lots of graphical analogies. It made the concept of functions limit (among other things) easily understandable. Unfortunately the said courses are copyrighted and closed source.


 * That's a great suggestion! If you have images or visualizations that you would like to donate, or if you know where we can find free or open-source images that we can use, that would be great. Unfortunately, creating lots of images takes time, and we are all just volunteers here. If you would like to join our project and help out, it would be appreciated. --Whiteknight (talk) (projects) 21:24, 10 January 2007 (UTC)


 * Do you mean adding images to (for example) the Sqrtx216.gif function? There's a link to an image link there - I can't decide whether it is a old link that's been deleted or a request for a new link. DavidMcKenzie 15:40, 6 February 2007 (UTC)

Shouldn't we start with the values approaching a particular value? That makes it a lot easier to understand.

Conversation
For what it matters, after learning a little more about formal continuity, density isn't even required. For instance, every function on the integers is continuous (since as a subspace of R, they have the discrete topology). Also, something interesting for anyone reading the person down there below Eric119: the integers are complete. Mo Anabre 20:59, 1 November 2007 (UTC)

I changed it back. The density of the rationals in the reals is quite enough to justify a concept of continuity of f. Mo Anabre 17:48, 1 May 2007 (UTC)

I don't understand why this is wrong. If the domain of f is only the rationals, then, according to our definition of limits (which seems to have vanished from the article), no matter what delta we have there will exist irrational x such that 0 < |x - c| < delta and thus f(x) is not defined and therefore the condition |f(x) - L| < epsilon cannot be true. (And then there is no limit and therefore a discontinuity.) If there is a problem in all this, please let me know. Eric119 01:28, 6 Sep 2004 (UTC)

this is just wrong! The real number line is complete, meaning there are no holes. However, if we say f (x) = 2 and restrict the domain of f to rational numbers, there will be holes at the places corresponding to irrational numbers. There are irrational numbers between any two rational numbers, so all the rational numbers on which f is defined are isolated from each other. But, there are rational numbers between any two irrational numbers, so each hole must be at a single point. There are infinite irrational numbers over any interval, so there are an infinite number of infinitely small holes between any two points on the graph. The places where f is defined are also infinite and infinitely small. The function f has no limits defined anywhere, despite that any graph would look like a simple line.

application to calculus out of context
The "application to calculus" section references an example and several figures that don't exist. I presume that the section was copied over from some other source, and either the context and figures were neglected, or they were deemed a copyright violation and deleted (but the "application to calculus" section remained). Please rewrite this section so that it no longer references material that doesn't exist on wikibooks! Thanks! Luvcraft 17:26, 2 July 2007 (UTC)

application to calculus out of context another note
I think - that limit itself has no too much "independent" uses... Actually - it is instrument for establishment of strict differentiation theory... So - I would just abandon this item... Good illustration of use of limit concept could be concept of tangent line, or "moment velocity" vector... But - they are once more - from field of physics and geometry, and not - pure calculus... --Simonovsky Pavel 18:29, 10 October 2007 (UTC)

As x ->0 sin (1/x) ->0
Sin (1/x) has no limit as x approaches 0. I have sources. The chapter must be changed. Zginder 13:43, 12 September 2007 (UTC)

Zginder, did you even read the section? I'm putting it back and fixing the _one_ error you didn't like. In the future please try to fix typos before deleting whole sections because of them. Mo Anabre 20:53, 1 November 2007 (UTC)

Problem with 'Limit rules'
Why are all the limit rules listed with no proofs? Shouldn't the be proven before they are given? 142.151.186.71 04:38, 9 October 2007 (UTC)
 * Agreed. Maybe the proofs can be put in an appendix with a working link from the 'Limit Rules' section or at the very end of the module. --Hamsterpoop 05:21, 29 October 2007 (UTC)
 * If it get some time here I'll try to come up with some of them, shouldn't bee too hard. 142.151.186.71 20:41, 3 November 2007 (UTC)

Began Edit
I started editing the beginning of the page, trying to start the discussion of a limit in a way that I (modestly) felt was more engaging, and tried to integrate it with the rest of the text. I hope to come back and try to do more work.

Thanks.

128.208.76.146 (talk)

Formal Definition of a Limit
Was the formal definition of a limit completely removed, or moved somewhere else? If the former, why? RSiferd (talk) 05:11, 21 May 2008 (UTC)
 * It appears later in this "chapter" Calculus/Formal Definition of the Limit. And again, almost identically, at Calculus/Formal limits. --Mrwojo (talk) 19:22, 28 December 2008 (UTC)
 * I've merged them at Calculus/Formal Definition of the Limit. --Mrwojo (talk) 05:54, 13 January 2009 (UTC)

Video?
Does wikibooks have embedvideo extension installed?

It does not appear to. Should it? Or some other video extension? Can we request this?

Introduction to Limits Lecture

~serundeputy

Infinite oscillation is redundant
The section on infinite oscillation is very redundant. Maybe it could be shortened? 173.74.178.117 (talk) 01:29, 7 July 2010 (UTC)


 * Thanks for the comment. There are not many people working on the book right now, so it could be a few years before someone acts on this comment.  But you can feel free to be bold and remove any parts that you don't like.  Don't worry, people quickly look over things that change because of the whole "watchlist" setup, so even if your just learning calculus yourself or an old pro you can still be an equally valid contributor.  You can do so with or without an account, but I usually recommend having one, so you can configure your preferences. Thenub314 (talk) 16:00, 7 July 2010 (UTC)

About mentioning L'Hôspital's rule with example 6
L'Hôspital's rule was briefly mentioned for finding the $$\lim_{x\to0}\frac{\sin(x)}{x}$$. However, this is technically mathematically incorrect because it would require you to know $$\frac{d}{dx}\left[\sin(x)\right]$$. This limit is something one would find when trying to prove the derivative of $$\sin(x)$$, so this means using L'Hôspital's rule is incorrect for this particular limit. I was wondering if I should delete the mentioning of L'Hôspital's rule and make a dedicated page proving "the limit." I am asking here to be sure no one minds this. Musical Inquisit (discuss • contribs) 03:45, 13 July 2020 (UTC)