Talk:One-sided limit

message: 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)
thread_title:  microhitlers 

message: You say:
The right-sided limit can be rigorously defined as:
\forall\varepsilon > 0\;\exists \delta >0 \;\forall x \in I \;(0 < x - a < \delta \Rightarrow |f(x) - L|<\varepsilon)
Similarly, the left-sided limit can be rigorously defined as:
\forall\varepsilon > 0\;\exists \delta >0 \;\forall x \in I \;(0 < a - x < \delta \Rightarrow |f(x) - L|<\varepsilon)
Where I  represents some interval that is within the domain of f
This appears to be nonsense.
Presumably it should read something like:
A real number L or one of the extended real numbers \infty, -\infty can be rigorously defined as a right-sided limit of fat a\in L(D \cap (a,\infty)) if respectively:
\forall\varepsilon\in\reals^+\;\exists \delta\in\reals^+\;\forall x \in D \;(0 < x - a < \delta \Rightarrow |f(x) - L|<\varepsilon)
\forall y\in\reals\;\exists \delta\in\reals^+ \;\forall x \in D \;(0 < x - a < \delta \Rightarrow f(x)>y) 
or
\forall y\in\reals\;\exists \delta\in\reals^+ \;\forall x \in D \;(0 <  x - a < \delta \Rightarrow f(x)<y)
Similarly, a real number L or one of the extended real numbers \infty, -\infty can be rigorously defined as a left-sided limit of fat a\in L(D\cap (-\infty ,a)) if respectively:
\forall\varepsilon\in\reals^+\;\exists \delta\in\reals^+\;\forall x \in D \;(0 < a - x < \delta \Rightarrow |f(x) - L|<\varepsilon)
\forall y\in\reals\;\exists \delta\in\reals^+\;\forall x \in D \;(0 < a - x < \delta \Rightarrow f(x)>y) 
or
\forall y\in\reals\;\exists \delta\in\reals^+\;\forall x \in D \;(0 <  a - x < \delta \Rightarrow f(x)<y)
Where D is the domain of f, L(S) denotes the set of limit points of a set S and \reals^+ denotes the set of positive real numbers.
The set of left(right)-sided limits of f at a 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 a\notin L(D \cap (a,\infty)) the right limit is not defined, and for a\notin L(D \cap (-\infty ,a)) 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 I, when the clause \forall x\in I\ldots becomes vacuously true). We would have, for example,
\lim_{x\to y^+}(0)=1\quad\quad \text{for all } y. 
Also if 
f(x)=0\quad x\in\mathbb{Q}
f(x)=1\quad x\notin\mathbb{Q}
then we could say that f(x) is continuous from the left and right throughout the real line. 
Without the restriction of a to L(D \cap (a,\infty)) or L(D\cap (-\infty ,a)) if D 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)
thread_title:  Definitions 

message: You say:
The two one-sided limits exist and are equal if the limit of f(x) as x approaches a exists.
This would not be necessarily true according to the definition I suggested in the previous section, e.g.:
\lim_{x \to 1}Arcsin(x)=\frac{\pi}{2}, 
but I have specified that 
\lim_{x \to 1^+}Arcsin(x) 
is undefined because 1\notin L(dom(Arcsin)\cap (1,\infty)). 
On what grounds do you assert this? — Preceding unsigned comment added by Martin Rattigan (talk • contribs) 05:13, 28 April 2015 (UTC)
thread_title:  Existence 

message: 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)
thread_title:  "rigorously defined" needs a link to syntax definition 

message: "In probability theory it is common to use the short notation: f(x-) for the left limit and f(x+) 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)
thread_title:  In probability theory?