Calculus I · Limits and Continuity · lesson
What a Limit Means
Learn What a Limit Means with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.
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 What a Limit Means; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Why Limits Matter in Calculus to Limits at Holes and Undefined Points. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Interpret in words; distinguish the input being approached from the output being approached; estimate a limit numerically and graphically.
The Basic Language of Limits
When we say "the limit is 7," we are not saying the input becomes 7. We are saying the output approaches 7 while the input approaches some specified number.
We write
when the values of can be made as close as desired to by taking sufficiently close to , with .
Read
as
"The limit of as approaches is ."
The roles are:
A limit with no hole
Let . What happens to as ?
Show worked solution
If is close to , then is close to :
| x | x+2 |
|---|---|
| 2.9 | 4.9 |
| 2.99 | 4.99 |
| 3.01 | 5.01 |
| 3.1 | 5.1 |
Therefore,
This limit is direct because the function has no break at . Later we will call this continuity.
limit-continuous-01Evaluate .
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Show hint
This polynomial has no break at .
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Source & rights
Original instruction with traceable references.
The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.
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