If a limit exists but the function does not equal it

coooool222 said:
View attachment 34427
Wouldn't this be false, since a limit can exist without the function being defined.
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I don't agree with your logic. If the function is not defined, then the limit will NOT exist. For a limit to exist you need a defined function!
What you meant to say is that a limit can exist at x=a without the function being defined at x=a.
For a limit to exist at x=a, the function MUST be defined around x=a, possibly not at x=a itself.
 
Steven G said:
I don't agree with your logic. If the function is not defined, then the limit will NOT exist. For a limit to exist you need a defined function!
What you meant to say is that a limit can exist at x=a without the function being defined at x=a.
For a limit to exist at x=a, the function MUST be defined around x=a, possibly not at x=a itself.
Click to expand...
@Steven G I do not agree that the student’s logic is defective. We can perhaps say that the problem’s presentation is sloppy because it implies that f(x) is defined over some relevant interval, but does not specify what that interval is. A sloppy problem will not elicit a precise answer.
 
Going on a tangent here, so feel free to ignore:

Lemma: if [imath]\lim_{x\rightarrow a} f(x)[/imath] exists than at least one of the statements is true:
A. [imath]f(a)[/imath] is defined, or
B. [imath]f(x)[/imath] is defined in the infinite number of points in any neighborhood of [imath]a[/imath].
 
blamocur said:
Going on a tangent here, so feel free to ignore:

Lemma: if [imath]\lim_{x\rightarrow a} f(x)[/imath] exists than at least one of the statements is true:
A. [imath]f(a)[/imath] is defined, or
B. [imath]f(x)[/imath] is defined in the infinite number of points in any neighborhood of [imath]a[/imath].
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Yes, I doubt the student formulated any such lemma, but assumed an inchoate version unconsciously. That is why I do not think the student had an error in logic. The student did not give a complete justification, but the wording of the problem was unlikely to elicit the exact justification correctly provided by Steven G.
 
JeffM said:
@Steven G I do not agree that the student’s logic is defective. We can perhaps say that the problem’s presentation is sloppy because it implies that f(x) is defined over some relevant interval, but does not specify what that interval is. A sloppy problem will not elicit a precise answer.
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Sorry @JeffM, but I still stand by my response about the student's logic flawed. Please re-read both posts from both the OP and me.
 
Steven G said:
I don't agree with your logic. If the function is not defined, then the limit will NOT exist. For a limit to exist you need a defined function!
What you meant to say is that a limit can exist at x=a without the function being defined at x=a.
For a limit to exist at x=a, the function MUST be defined around x=a, possibly not at x=a itself.
Click to expand...
It seems obvious to me what the student meant, which is just what you said:
coooool222 said:
... a limit can exist without the function being defined.
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Obviously the statement refers to a particular point, as that is required in order to talk about "a limit" at all, and the question was about a limit at a particular point. So clearly the meaning is

a limit can exist [at a given point] without the function being defined [at that point].​

It's fine to point out that a student's wording can be improved, but there's no need to tell them that their thinking itself is wrong, when it clearly isn't. I greatly prefer to correct a student in a positive way, pointing out first that they are right, and then suggesting how I would say it.

After all, we aren't the teacher grading a proof; we are letting them informally tell us their thinking. And our goal isn't to drive students away from math.
 
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