Скачать или смотреть Calculus 1 — 5.5: One-Sided Limits and Vertical Asymptotes

A two-sided limit only exists when the function approaches the same value from both directions. This lesson introduces left-sided and right-sided limit notation, shows how to evaluate one-sided limits using tables and piecewise functions, and explains what happens when outputs grow without bound near a vertical asymptote.

Key concepts covered:
• Right-sided limit notation lim x→a⁺ (approaching from values greater than a) and left-sided limit notation lim x→a⁻ (approaching from values less than a)
• The core theorem: lim x→a f(x) = L if and only if both one-sided limits equal L
• Limits describe approach, not arrival — f(a) is irrelevant
• Worked example: piecewise function where left limit equals −1 and right limit equals 1, so the two-sided limit does not exist
• Infinite limits with f(x) = 1/x as x approaches 0 — right side heads to +∞, left side heads to −∞
• Definition of a vertical asymptote: a line x = a where one or both one-sided limits are infinite
• Four possible one-sided behaviors near a vertical asymptote (±∞ from each side)
• Comparing 1/x (sides disagree, limit DNE) with 1/x² (both sides head to +∞, limit equals +∞)
• Practice: reading one-sided limits from four graphs to determine if the two-sided limit exists
• Decision flowchart: both sides match a finite L → limit is L; both sides match the same infinity → limit is that infinity; sides disagree → limit DNE

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SOURCE MATERIALS
The source materials for this video are from    • Calculus 1 Lecture 1.1:  An Introduction t...