Calculus 1 — 6.4: Piecewise Function Limits скачать в хорошем качестве

Calculus 1 — 6.4: Piecewise Function Limits

Learn the systematic method for evaluating limits of piecewise functions by analyzing boundary points. Instead of checking every x-value, you only need to examine where the pieces meet — comparing left-hand and right-hand limits using direct substitution into the adjacent formulas. Key concepts covered: • Piecewise functions as multiple formulas stitched across intervals • Why only boundary points (where pieces meet) require limit analysis • One-sided limits: left-hand and right-hand limit computation • The fundamental test: a limit exists at a boundary if and only if both one-sided limits are equal • Identifying which piece governs each side of a boundary point • Direct substitution for continuous pieces (polynomials, square roots) • Handling 0/0 indeterminate forms by factoring and canceling (removable discontinuities) • Recognizing nonzero-over-zero results as vertical asymptotes (non-removable discontinuities) • Sign analysis to determine whether a limit tends toward positive or negative infinity • Full worked example: f(x) = x+1 for x less than -2, x²-5 for -2 ≤ x less than 3, and √(x+1) for x ≥ 3 • Step-by-step boundary checks at x = -2 (limit exists, equals -1) and x = 3 (limit DNE, one-sided values 4 ≠ 2) • Three-step flowchart: identify boundaries, compute one-sided limits, compare values ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from    • Calculus 1 Lecture 1.2:  Properties of Lim...