7bcd623519e9200ebd21081 20260605012231491 hyperscienceij@gmail.com:rcrl hyperscienceij@gmail.com WEB-FORM Hyperscience International Journals hij 28213300 06 2026 6 2 James Russell Farmer Independent Researcher, 10 William Ave, Greenlane, Auckland 1051, New Zealand https://orcid.org/0000-0001-8131-8835 The inverse function occupies a special position in elementary calculus because the standard power-law rule for integration ‎fails for a particular exponent. This exceptional behavior motivates a broader investigation into the mathematical structure of ‎logarithmic and hyperbolic functions. In this article we examine the asymptotic behavior of logarithmic and inverse functions, ‎with particular emphasis on singularities, asymptotes, and intersection structure. We investigate how logarithmic behavior ‎naturally emerges from inverse power relationships and discuss the geometric significance of the transition between these two ‎classes of functions. The discussion is then extended toward conceptual analogies in physics, especially in relation to ‎asymptotic behavior appearing in special relativity and gravitational theory. Rather than proposing modifications to ‎established physical theories, the article explores mathematical parallels between divergent structures, limiting processes, and ‎physical interpretation.‎ The aim of this work is therefore not to replace existing physical theories, but to highlight mathematical patterns that recur in ‎both analysis and theoretical physics.‎ 06 2026 19 24 10.55672/hij2026pp19-24‎ https://hscience.org/index.php/hij/article/view/197