Calculus I · Limits and Continuity · lesson
Limits at Holes and Undefined Points
Learn Limits at Holes and Undefined Points with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practi
Where this chapter fits
Chapter 1: What a limit means
Build the neighborhood idea with motion, tables, holes, one-sided behavior, and graphs.
Reading lens: What are nearby outputs doing as the input approaches the target from both sides? Keep that question in view while reading Limits at Holes and Undefined Points; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects What a Limit Means to Function Value vs. Limit. Read the explanation first, predict each example’s next move, and only then compare the written solution.
A function can have a limit where it is undefined
Consider
At , the formula is undefined. For , however,
So the graph is the line with one point removed.
| 1.9 | 3.9 |
| 1.99 | 3.99 |
| 1.999 | 3.999 |
| 2.001 | 4.001 |
| 2.01 | 4.01 |
| 2.1 | 4.1 |
Both sides approach , so
The function is undefined at , but nearby outputs approach .
To estimate a limit from a table, use inputs on both sides of the target. The table should move progressively closer to the target. A table gives evidence, not proof; algebra or a trustworthy graph is usually needed to confirm the pattern.
Table, graph, and algebra agree
Estimate and then evaluate
Show worked solution
Numerical evidence.
| x | \dfrac{x^3-1}{x-1} |
|---|---|
| 0.9 | 2.71 |
| 0.99 | 2.9701 |
| 0.999 | 2.997001 |
| 1.001 | 3.003001 |
| 1.01 | 3.0301 |
| 1.1 | 3.31 |
The values appear to approach .
Algebraic confirmation. Use the difference-of-cubes identity:
For ,
Therefore,
Source & rights
Original instruction with traceable references.
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