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I removed the following example:

  • We have
whereas

As far as I know, the arctangent is continuous everywhere on the real line, and this limit is zero. I would like to replace the example with one that does work, but I don't know any from trigonometry. It is true that lim x-> infty arctan x is pi/2, but one sided limits aren't very enlightening at infinity. If someone would like to explain what was supposed to be there, I'd like to have it back. -lethe talk 05:11, 18 November 2005 (UTC)

I've put it back with the needed correction. It now says the following:

  • We have
whereas

Michael Hardy 23:48, 18 November 2005 (UTC)[reply]

Now that you've said what it was supposed to be, it should have been obvious. Anyway, thanks. -lethe talk 00:10, 19 November 2005 (UTC)
I found it a bit strange on Michael's behalf to remove an elementary example with a picture, in favor of an example which duplicates an existing one, that is containing 1/x with x->0.
By no means do I plan it as a revenge, but after Lethe put my example back, I removed Michael's second example. Now we have two very different and somewhat representative examples, as things should be. Oleg Alexandrov (talk) 00:38, 19 November 2005 (UTC)[reply]

microhitlers

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The following sentance does not seem appropriate to me but i'm not familiar with this subject so I don't want to rush in and make corrections

"The f(x±) notation is singularly and uniformly awful (600 microHitlers)"

Plugwash (talk) 19:45, 9 February 2009 (UTC)[reply]

Definitions

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You say:

The right-sided limit can be rigorously defined as:

Similarly, the left-sided limit can be rigorously defined as:

Where represents some interval that is within the domain of

This appears to be nonsense. Presumably it should read something like:

A real number or one of the extended real numbers , can be rigorously defined as a right-sided limit of at if respectively:

or


Similarly, a real number or one of the extended real numbers , can be rigorously defined as a left-sided limit of at if respectively:

or


Where is the domain of , denotes the set of limit points of a set and denotes the set of positive real numbers.

The set of left(right)-sided limits of at contain at most one element which, if it exists, is the left(right)-sided limit and is denoted as above. If either set is empty the corresponding limit is not defined. For the right limit is not defined, and for the left limit is not defined.

As it stands all real numbers would be left(right)-sided limits of all functions at all points (one need only choose the empty interval for , when the clause becomes vacuously true). We would have, for example,

.


Also if

then we could say that is continuous from the left and right throughout the real line.


Without the restriction of to or if contained points isolated from below or above then all reals would also be left and/or right limits at these points. Martin Rattigan (talk) 02:04, 28 April 2015 (UTC)[reply]

Existence

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You say:

The two one-sided limits exist and are equal if the limit of as approaches exists.

This would not be necessarily true according to the definition I suggested in the previous section, e.g.:

,


but I have specified that

is undefined because .


On what grounds do you assert this? — Preceding unsigned comment added by Martin Rattigan (talkcontribs) 05:13, 28 April 2015 (UTC)[reply]

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I love that there is a rigorous definition tabled. I suggest that for the lay reader interested in learning more about that there is no easy or obvious way to do that. It's a classic case of "what to search for". Does one search "rigorous definitions in math" for example and wade through a lot of results in the hope of finding a page that describes how to read this? I think not. The beauty of the web, and of wikipedia is that we can link directly from the words "rigorously defined" or via a parenthesized or "see also" link point readers to a Wikipedia page that describes the syntax of this rigorous definition. As I am such a lay reader with no clue where to turn, I can't even add it, just drop a note here saying how nice it would be: if assumed knowledge were low and links to learning more are embedded. --120.29.241.112 (talk) 03:12, 15 June 2018 (UTC)[reply]

In probability theory?

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"In probability theory it is common to use the short notation: for the left limit and for the right limit." — Really? I always believed this is common in math analysis (and all theories that use it, including probability). Boris Tsirelson (talk) 04:38, 24 September 2019 (UTC)[reply]