Interactive Lab / Continuity
Diagnose how the function fails.
Move toward the suspect point from both sides. The scanner keeps the point value, one-sided samples, limiting behavior, and repairability separate.
Two-Sided Inspection
f(x) = (x² - 1)/(x - 1), x ≠ 1
From the leftx = 0.99f(x) = 1.99
From the rightx = 1.01f(x) = 2.01
Value at pointUndefined
Two-sided verdictFinite limit: 2
ClassificationRemovable
Both sides approach 2. Defining f(1)=2 repairs continuity.
Comparison Record
| Model | h | Left sample | Right sample | Verdict |
|---|
Oscillation needs witness sequences
For sin(1/x), ordinary decimal samples can accidentally suggest a pattern. The analytical diagnosis uses sequences approaching zero that force the function to equal 1 and -1 repeatedly. Because those outputs remain separated while both input sequences approach the same point, no limit exists.