Variations and extensions of Croft's problem

TL;DR

This paper investigates a classical calculus problem concerning the behavior of continuous functions and the conditions under which their limits at infinity can be inferred from the limits of their scaled sequences.

Contribution

It explores variations and extensions of Croft's problem, providing new insights and potential solutions to this longstanding mathematical challenge.

Findings

Identifies conditions under which the limit behavior of functions can be deduced from scaled sequences.

Proposes new approaches to analyze the problem's variations.

Offers partial solutions or conjectures for specific classes of functions.

Abstract

In this work we study the following classical still challenging Calculus problem: {\it If $f:(0,\infty)\to\mathbb{R}$ is a continuous function, for which the sequence $\{f(nx)\}$ tends to zero, for every positive $x$, as $n$ tends to infinity, then $f(x)$ also tends to zero, as $x$ tends to infinity.}