Math in the Modern World: Mastering Piecewise Limits and Indeterminate Forms

This specialized guide deep-dives into the critical distinction between a function’s limit and its actual value, focusing on the common "traps" found in calculus-prep courses. It provides an exhaustive breakdown of how to solve limits when $f(c)$ is undefined, specifically addressing the Indeterminate Form (0/0) through algebraic factoring techniques. The document features detailed walkthroughs of Piecewise Functions, demonstrating step-by-step how to calculate left-hand vs. right-hand limits to prove existence or identify jump discontinuities. With a mix of coordinate geometry, limit notation, and table-based estimation, this resource is designed to help students navigate the exact types of "non-continuous" problems that frequently appear on midterms and finals.