Talk:One-sided limit

Definitions

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 $f$at $a\in L(D\cap (a,\infty ))$ if respectively:

$\forall \varepsilon \in \mathbb {R} ^{+}\;\exists \delta \in \mathbb {R} ^{+}\;\forall x\in D\;(0<x-a<\delta \Rightarrow |f(x)-L|<\varepsilon )$

$\forall y\in \mathbb {R} \;\exists \delta \in \mathbb {R} ^{+}\;\forall x\in D\;(0<x-a<\delta \Rightarrow f(x)>y)$

or

$\forall y\in \mathbb {R} \;\exists \delta \in \mathbb {R} ^{+}\;\forall x\in D\;(0<x-a<\delta \Rightarrow f(x)<y)$

Similarly, a real number $L$ or one of the extended real numbers $\infty$, $-\infty$ can be rigorously defined as a left-sided limit of $f$at $a\in L(D\cap (-\infty ,a))$ if respectively:

$\forall \varepsilon \in \mathbb {R} ^{+}\;\exists \delta \in \mathbb {R} ^{+}\;\forall x\in D\;(0<a-x<\delta \Rightarrow |f(x)-L|<\varepsilon )$

$\forall y\in \mathbb {R} \;\exists \delta \in \mathbb {R} ^{+}\;\forall x\in D\;(0<a-x<\delta \Rightarrow f(x)>y)$

or

$\forall y\in \mathbb {R} \;\exists \delta \in \mathbb {R} ^{+}\;\forall x\in D\;(0<a-x<\delta \Rightarrow f(x)<y)$

Where $D$ is the domain of $f$, $L(S)$ denotes the set of limit points of a set $S$ and $\mathbb {R} ^{+}$ 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)

Keeping the [previous], and adding an issue with the definitions more recently. In general definitions should be precise, or simply, not stated.

"If I represents some interval" and "Let I represent an interval where $I \subseteq \mathsf{domain}(f)$..."

Neither of those statements state what I is, nor how it is bound in the definition. For example, it is not stated whether the interval is open or closed! This is just a kind of general sloppiness, but it can easily turn into gatekeeping under "you should know what I mean." Presumably, "open" interval is meant.

Also, by not specifying if the interval needs to begin or end with "a", it seems one can get into trouble with $\forall x \in I. 0 < a-x ...$. For example for the left-limit, sure, if $0< a-x$ fails, then ... it's technically, fine because we're only going to consider the implication for when it's true. However, one then needs that the interval I has some interior on the points: $\{z \in I \, | \, z < a\} \ne \emptyset$. However, the moment that is the case we can use $(b,a)$ for any $b< a$, but we can, unless I'm mistaken use the ray $(-\infty,a)$. It seems WLOG then one could drastically simplify the "symbolic" version as $\lim_{x\to a^{-}}f(x) = L$ precisely when for all $\epsilon >0$, there exists a $\delta >0$ such that for all $x < a$, whenever $a-x < \delta$ we have $|f(x)-L| < \epsilon$.

This makes it absolutely clear it's just for $x< a$. To use the interval, one would seem to need to assume that there exists an interval $(b,a)$ with $b < a$ and then instead of for all $x< a$ we could do for all $x\in I$. But this seems clumsy to me, for no gain. But again, I'm not really sure what the goal here is. — Preceding unsigned comment added by 192.136.116.2 (talk) 02:14, 12 November 2024 (UTC)

Existence

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)

"rigorously defined" needs a link to syntax definition

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)

In probability theory?

"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)