Mean Value Theorems and L'Hospital-Type Rules for Regulated Functions

2023/05/27 by Ahmed Ghatasheh, Ghatasheh, Ahmed
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #History and Overview (math.HO) #Mathematical Dynamics and Fractals #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2305.17572

openalex publication_date 2023/05/27 · openalex created_date 2023/05/31 · openalex updated_date 2026/07/28

Abstract

We introduce a generalization of Cauchy's mean value theorem for regulated functions. Building on this, we extend both L'Hospital's rule and L'Hospital's monotone rule to quotients of regulated functions. We demonstrate that our extended L'Hospital's rule encompasses both the discrete case, known as the Stolz-Cesaro theorem, and the classical continuous case. In addition, we show that these extensions handle some problems that classical rules cannot address. Finally, we provide Lebesgue-Stieltjes versions of L'Hospital's rule and L'Hospital's monotone rule and compare them with our extensions.

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