Exploring Limits: Understanding the Existence of lim x->a f(x)

[Puky]

Hello,

Say f(x) is defined only for x in [a, ∞].
lim x->a⁺ f(x) = c and
lim x->a^- f(x) obviously doesn't exist.
Do we say that lim x->a f(x) exists or not?

Thanks.

 

[arildno]

What do you think?

 

[Puky]

Not sure. If the function is defined only for, say, x in [a, ∞], would we say the limit at a doesn't exist? Sorry, I forgot to add that to the OP.

 

[arildno]

Not sure. If the function is defined only for, say, x in [a, ∞], would we say the limit at a doesn't exist? Sorry, I forgot to add that to the OP.

An excellent objection!
What you show with your objection is the concern that whether or not a limit exists can depend, in a CRUCIAL way, on what the domain the variable is said to "live in".
IF, as you you object, x only lives in the region between "a" and positive infinity, then the limit most definitely does exist (because then, only the a+ limit is meaningful to apply at "a"!).

However, if x is conceived as living along the whole number line, then the limit does NOT exist.

So, in general, both you (and the textbook authors!) have to be clear about what is the actual DOMAIN your variable lives on.

Once THAT is clear, then your conclusion concerning the limit can be made in a definite, and clear, manner.

 

[arildno]

When a function is defined at only a restricted interval, it is meaningless to speculate how it behaves outside that interval, and hence, whatever the values there, they have no relevance for the resolution of the question whether the limit of the function exists at some point or not.

you say:
"lim x->a- f(x) obviously doesn't exist."
Nope.
It is a MEANINGLESS assertion, in this particular context, since you can¨t use the lim operation where no x's exist to evaluate it.
The limit neither( exists) or (does not exist), from THAT direction.

 

[Puky]

An excellent objection!
What you show with your objection is the concern that whether or not a limit exists can depend, in a CRUCIAL way, on what the domain the variable is said to "live in".
IF, as you you object, x only lives in the region between "a" and positive infinity, then the limit most definitely does exist (because then, only the a+ limit is meaningful to apply at "a"!).

However, if x is conceived as living along the whole number line, then the limit does NOT exist.

So, in general, both you (and the textbook authors!) have to be clear about what is the actual DOMAIN your variable lives on.

Once THAT is clear, then your conclusion concerning the limit can be made in a definite, and clear, manner.

but:
Since your question has already allowed for the the existence of x's less than "a", what should therefore be your conclusion?

Thank you very much for your answer, that is what I was thinking. I asked because I remember running into these kinds of questions a few times, in fact right now I'm looking at a textbook that says lim x->0 $\sqrt{x^3-x}$ doesn't exist because the right-hand side limit doesn't exist, without specifying the domain. I just don't want to lose points in exams for silly reasons

 

[arildno]

In that specific case you mention , you should point out that lim x->0 only is meaningful for values of x less than zero (and greater than x=-1), and that therefore, the only x-values you can judge the limit by necessitates that the conclusion that the limit at x=0 exists.
The existence of a limit at some point requires that you have the ability to form convergent sequences WITHIN the domain to that point, in order to evaluate whether or not the limit exists.
If that ability is lacking due to "faults" in how the domain can be constructed, then that is a fault of the domain, not the fault in the limit.

 

[Puky]

I've got it now, thank you very much, your answers were really helpful.