---
OA_place: repository
OA_type: green
_id: '21737'
abstract:
- lang: eng
  text: 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.
acknowledgement: "This project was funded in part by the European Research Council
  (ERC Consolidator Grant 101045083 CoDiNA) and the National Science Foundation CAREER
  Award 2239062.\r\n"
article_number: '2602.09958'
article_processing_charge: No
arxiv: 1
author:
- first_name: Albert
  full_name: Chern, Albert
  last_name: Chern
- first_name: Sadashige
  full_name: Ishida, Sadashige
  id: 6F7C4B96-A8E9-11E9-A7CA-09ECE5697425
  last_name: Ishida
  orcid: 0000-0002-3121-3100
citation:
  ama: Chern A, Ishida S. L’Hopital rules for complex-valued functions in higher dimensions.
    <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/ARXIV.2602.09958">10.48550/ARXIV.2602.09958</a>
  apa: Chern, A., &#38; Ishida, S. (n.d.). L’Hopital rules for complex-valued functions
    in higher dimensions. <i>arXiv</i>. <a href="https://doi.org/10.48550/ARXIV.2602.09958">https://doi.org/10.48550/ARXIV.2602.09958</a>
  chicago: Chern, Albert, and Sadashige Ishida. “L’Hopital Rules for Complex-Valued
    Functions in Higher Dimensions.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/ARXIV.2602.09958">https://doi.org/10.48550/ARXIV.2602.09958</a>.
  ieee: A. Chern and S. Ishida, “L’Hopital rules for complex-valued functions in higher
    dimensions,” <i>arXiv</i>. .
  ista: Chern A, Ishida S. L’Hopital rules for complex-valued functions in higher
    dimensions. arXiv, 2602.09958.
  mla: Chern, Albert, and Sadashige Ishida. “L’Hopital Rules for Complex-Valued Functions
    in Higher Dimensions.” <i>ArXiv</i>, 2602.09958, doi:<a href="https://doi.org/10.48550/ARXIV.2602.09958">10.48550/ARXIV.2602.09958</a>.
  short: A. Chern, S. Ishida, ArXiv (n.d.).
corr_author: '1'
date_created: 2026-04-15T16:28:24Z
date_published: 2026-02-10T00:00:00Z
date_updated: 2026-04-28T10:56:30Z
day: '10'
ddc:
- '510'
department:
- _id: GradSch
- _id: ChWo
doi: 10.48550/ARXIV.2602.09958
external_id:
  arxiv:
  - '2602.09958'
file:
- access_level: open_access
  checksum: 6a76591c723d3e949ad5afa9f7dbb2ee
  content_type: application/pdf
  creator: dernst
  date_created: 2026-04-28T10:53:27Z
  date_updated: 2026-04-28T10:53:27Z
  file_id: '21771'
  file_name: 2026_arXiv_2602.09958.pdf
  file_size: 867109
  relation: main_file
  success: 1
file_date_updated: 2026-04-28T10:53:27Z
has_accepted_license: '1'
keyword:
- l’Hopital theorem
- complex functions
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '02'
oa: 1
oa_version: Preprint
project:
- _id: 34bc2376-11ca-11ed-8bc3-9a3b3961a088
  grant_number: '101045083'
  name: Computational Discovery of Numerical Algorithms for Animation and Simulation
    of Natural Phenomena
publication: arXiv
publication_status: submitted
status: public
title: L'Hopital rules for complex-valued functions in higher dimensions
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2026'
...