AP Calculus AB and BC
Limit Does Not Exist vs Undefined
A function being undefined at a point says nothing about its limit there, because the limit only depends on nearby values and never on the point itself. A removable discontinuity is exactly the case where the function is undefined but the limit exists perfectly well.
Undefined at a
Use when: You are asked for the function VALUE, the domain, or whether the function is continuous there.
Limit does not exist
Use when: You are asked about the limit, which needs the one-sided limits to agree on a single finite value.
Side by side
| Undefined at | Limit does not exist | |
|---|---|---|
| About | The value | The behaviour NEAR |
| Can the other still hold | Yes, the limit may exist | Yes, may be defined |
| Example: at | Undefined | Limit exists and is |
| Example: at | Undefined | Does not exist, sides disagree |
| Continuity needs | Defined | Limit to exist AND equal the value |
The two examples are the whole page. Both functions are undefined at zero, yet one has a perfectly good limit and the other does not. Being undefined is a fact about a single point; a limit is a fact about a neighbourhood.
Continuity needs all three
f(a) must exist, the limit must exist, and they must be equal. Each of those can fail independently, which is why there are several distinct kinds of discontinuity rather than one.
Frequently asked questions
If a function is undefined at a point, does the limit exist?
It may. is undefined at and has limit . The limit never evaluates the point itself.
Can a limit fail to exist where the function IS defined?
Yes. A jump discontinuity has a defined value and disagreeing one-sided limits.
What does continuity require?
Three things: the value exists, the limit exists, and they are equal.
In the CED: Unit 1: Limits and Continuity