Rigorous Derivations of Exponential Limits and the Base of Natural Logarithms

The mathematical constant e plays a foundational role in real analysis, calculus, differential equations, and complex analysis. This paper provides formal, step-by-step mathematical derivations of two key limit expressions involving the exponential function: the general power-limit expression for the exponential function and its classic foundational base limit. By leveraging natural logarithmic transformations, continuity properties of elementary functions, and L'Hôpital's Rule for indeterminate forms, it demonstrates the analytical equivalence of these limits. Furthermore, alternative derivation methods including binomial expansions and logarithm derivative techniques are discussed to highlight structural insights across real analysis.