On Some Types of Limit

Ufuk Kaya; Gokhan Turan

Home/Archive/arXiv

arXiv · 2312.14272

On Some Types of Limit

Ufuk Kaya·Gokhan Turan

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we consider the concept of limit, one of the basic concepts of mathematical analysis. At a point $a\in{\mathbb{R}}$, the limit of a function $f$ from $A\subset\mathbb{R}$ to $\mathbb{R}$ is $L\in{\mathbb{R}}$ if and only if there exists $\delta>0$ such that the set $$ \left\{x\in\left(\left(a-\delta,a+\delta\right)\backslash\left\{a\right\}\right)\cap A:\lvert f\left(x\right)-L\rvert \ge\varepsilon\right\} $$ is the empty set for each $\varepsilon>0$. This study investigates what happens when the above set is not empty. More clearly, when we take the above set as finite or a set whose accumulation points is empty, we prove that the limit concept does not change. However, when we take the above set as countable or a set of measure zero, we prove that they are new limit concepts. Also, we show that these new limit concepts have the properties of existence, uniqueness, additivity, multiplicativity, etc. Finally, by giving some examples, we compare the limit concepts we have obtained with the previous ones.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Ufuk Kaya, Gokhan Turan. 2023-12-21. On Some Types of Limit. https://arxiv.org/abs/2312.14272

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗