Math in the Modern World - Solving Polynomial and Radical Limits with Substitution

This class notes provides a rigorous walkthrough of evaluating limits for complex algebraic functions, moving beyond basic estimation into advanced analytical methods. It begins with the Direct Substitution Property for polynomial and rational functions, establishing the foundational rule that the limit as $x$ approaches $c$ is simply $f(c)$, provided the denominator is non-zero. The guide then tackles high-level algebraic hurdles, such as simplifying expressions with radical denominators through rationalization and handling complex fractions. Featuring worked-out solutions for square root functions and quadratic ratios, this document is an essential asset for students needing to master the "Substitution Method" and conjugate multiplication to solve indeterminate forms in calculus.