The Recursive Growth-Rate Limit Method: A Unified Approach to Indeterminate Forms

We present a formal study of the recursive growth-rate limit method for evaluating limits of the form lim_{x→a} f(x)/g(x), which systematically handles all standard indeterminate forms. This method, grounded in the Growth Rate Principle inspired by Hilbert's Grand Hotel Paradox, provides a unified algorithmic approach to computing limits by repeatedly differentiating numerator and denominator until the quotient becomes determinate. We prove rigorously that non-quotient indeterminate forms (0·∞, ∞–∞, 0⁰, 1^∞, ∞⁰) can be algebraically or logarithmically converted to quotient forms 0/0 or ∞/∞, to which the recursive derivative method applies. We compare this technique to classical tools such as Taylor expansions, asymptotic (Big-O) analysis, and Hardy field theory. Finally, we discuss applications in mathematics (asymptotic estimates, series), physics (singular perturbation limits), computer science (algorithmic complexity), and modeling (growth processes).