Limits of functions at end points of its domain

Question

I was reading Thomas' calculus and got to know the following: -

for the function $\sqrt{4-x^2}$ clearly it has its domain $[-2,2]$ , function has a one sided limit only at $-2,2$ so $-2$ has only RHL and 2 has only LHL. so it says that at the end points the limit exists but it is one sided .
then it says in a theorem that at a boundary point of its domain , a function has a limit when it has an appropriate one sided limit.

My question is does appropriate here mean finite or does it mean some other thing? Thank you

It means the limit exists. For example, on $(0, 1] \lim_{x^+ \to 0} \ln(x) $ doesn't exist as $\ln(x)$ is not defined at $x = 0$. — layabout, Oct 10 '20 at 20:09
can it be plus minus infinite , the limit at a boundary point — chittaranjan rout, Oct 10 '20 at 20:17
sure, for example on $(0, 1] \lim_{x^+ \to 0} 1/x = \infty$. — layabout, Oct 10 '20 at 20:18
but, i read that the limit exists when lhl=rhl=some finite value an that value is the limit or is the case different for one sided limits — chittaranjan rout, Oct 10 '20 at 20:19
I guess that "appropriate" means here that you have to consider left hand limit if the point is on the right side, and right hand limit if the point is on the left side. — TheSilverDoe, Oct 10 '20 at 20:20
but, i read that the limit exists when lhl=rhl=some finite value an that value is the limit or is the case different for one sided limits — chittaranjan rout, Oct 10 '20 at 20:26
Please clear my concepts regarding limits. We read that limit exists at a point inside the domain if LHL=RHL=Some finite value. But if we see the boundary points it has only one sided limit. So its limit is that one sided limit . My doubt is that ' can the limit exist at a point "a" if the function's domain is(a,b) ? Next question is if we see a function which has "a" in its domain can its limit be +-infinity. What will be for the case "a" not in domain but domain is (a,b)? thank you. — chittaranjan rout, Oct 10 '20 at 21:03

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