L'Hopital rules for complex-valued functions in higher dimensions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911438252015616 |
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| author | Chern, Albert Ishida, Sadashige |
| author_facet | Chern, Albert Ishida, Sadashige |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09958 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | L'Hopital rules for complex-valued functions in higher dimensions Chern, Albert Ishida, Sadashige Complex Variables 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. |
| title | L'Hopital rules for complex-valued functions in higher dimensions |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2602.09958 |