On the Analytical Equivalence of Power-Limit Definitions and Maclaurin Expansion for Exponential Functions

The exponential function and its defining base constant serve as fundamental pillars across real analysis and applied mathematics. This paper presents formal, step-by-step mathematical derivations of the foundational power-limit expressions for both the general exponential function and its base constant. Utilizing logarithmic transformations, continuity properties of elementary functions, and L'Hôpital's Rule for indeterminate forms of the one-to-the-power-of-infinity type, it rigorously resolves these limit identities. Additionally, this paper integrates a Maclaurin series expansion framework to demonstrate the direct analytical equivalence between limit-based definitions and infinite power series, unifying differential calculus and series representations.