L'Hopital rules for complex-valued functions in higher dimensions

TL;DR

This paper extends l'Hopital's rule to complex-valued functions in higher dimensions, providing a complete characterization for when their quotients are continuous at points where both functions vanish.

Contribution

It offers a novel, comprehensive criterion for the continuity of quotients of complex functions in higher dimensions, advancing understanding beyond the real-valued case.

Findings

Complete characterization of quotient continuity in complex higher dimensions

Identification of subtleties absent in real-valued l'Hopital's rule

Discussion of challenges in extending to smoother quotients

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.