L'Hopital rules for complex-valued functions in higher dimensions
Complex Variables
2026-02-11 v1
Abstract
In calculus, l'Hopital's rule provides a simple way to evaluate the limits of quotient functions when both the numerator and denominator vanish. But what happens when we move beyond real functions on a real interval? In this article, we study when the quotient of two complex-valued functions in higher dimension can be defined continuously at the points where both functions vanish. Surprisingly, the answer is far subtler than in the real-valued setting. We provide a complete characterization for the continuity of the quotient function. We also point out why extending this result to smoother quotients remains an intriguing challenge.
Cite
@article{arxiv.2602.09958,
title = {L'Hopital rules for complex-valued functions in higher dimensions},
author = {Albert Chern and Sadashige Ishida},
journal= {arXiv preprint arXiv:2602.09958},
year = {2026}
}