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Limits
Distinguish the value at a point from the value approached nearby.
Start with the picture
Separating the point value from nearby behaviour reveals the distinction between limits and continuity.
Check with another example
1. If f(x)=x+2 for x≠2, what is the limit as x→2?
- 4
- No limit because the value at 2 is missing
- 0
My reason
2. For f(x)=1 to have a limit at 3, must its values differ from 1?
- Yes: a limit must never be reached
- No: it can already equal 1
- A constant is not a function
My reason
3. If f(x)=x+2 for x≠2 but f(2)=1, what is the limit at 2?
- 1
- There is no limit
- 4
My reason
Explain in your own words
How does checking three nearby inputs differ from proving a limit?
Answers and reasons
1. 4
The output error |f(x)−4| equals |x−2| nearby.
2. No: it can already equal 1
Yes. What matters is nearby behaviour, not whether the value is reached.
3. 4
The limit is 4 because f(x)=x+2 nearby. It differs from f(2)=1, so the function is not continuous there.
A table checks only those inputs. A proof must control all sufficiently close inputs for any prescribed positive tolerance.