Скачать или смотреть Calculus 1 — 2.4: Holes vs Vertical Asymptotes and Finding Range

Learn to distinguish between holes (removable discontinuities) and vertical asymptotes in rational functions using a simple diagnostic test — then apply those skills to finding range and solving applied domain problems.

When two rational functions share the same domain restriction, one might have a tiny missing point while the other explodes to infinity. This lesson shows you exactly how to tell which is which by plugging the restricted value into the original function and interpreting the result: 0/0 signals a hole, while a nonzero number over zero signals a vertical asymptote.

Key concepts covered:
• The 0/0 vs nonzero/0 diagnostic test for classifying domain restrictions
• Holes (removable discontinuities): factoring, canceling, and finding the missing point's coordinates
• Vertical asymptotes: why no cancellation can fix a nonzero/0 result
• Worked example: f(x) = (x² − 4)/(x − 2) has a hole at (2, 4)
• Worked example: g(x) = 12/(x − 4) has a vertical asymptote at x = 4
• Finding range by evaluating at domain boundaries (square root functions)
• Finding range by solving for x in terms of y (rational functions)
• Edge case: square roots in denominators require the radicand to be strictly positive
• Applied domain: the open-top box problem from a 16 × 30 inch sheet of cardboard
• Deriving the volume function V(x) = x(16 − 2x)(30 − 2x)
• Physical constraints narrowing the mathematical domain to the interval (0, 8)
• Identifying the binding constraint among multiple inequalities

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SOURCE MATERIALS
The source materials for this video are from    • Calculus 1 Lecture 0.2:  Introduction to F...