If you looks at this , it's kind of trivial. Once you find the inner limit, it'll be just a number or there won't be a limit. The outer limit is meaningless in this case.
However, there is the following
For this you need uniform convergence.
Is there a theorem on the composition of limits? What I mean is, is there some equation relating the limits of two functions with the limit of the composition of a function? And what about 'nested' limits? I know the following theorems about limits:
and
and
IF then
But what about something that would give a simplification of the following limits:
(Assuming that we know and , and assuming these limits exist)
Is there anything we can say about the limit of the composition of these functions? And what about nested limits? Is there anything we can say about an expression like:
? Is there any "rule" about nested limits that can be used to help in the manipulation of such an expression?
Thanks in advance
Be very careful with these "rules". They pretty much only apply when the limits on the right hand side exist and are finite. If you forget about that, you will get the wrong answers to questions later on.
Well, If f is continuous at , then , and similarly for the other thing.But what about something that would give a simplification of the following limits:
(Assuming that we know and , and assuming these limits exist)
See what Vlasev said (and if you haven't heard of uniform convergence, don't worry about it).Is there anything we can say about the limit of the composition of these functions? And what about nested limits? Is there anything we can say about an expression like:
? Is there any "rule" about nested limits that can be used to help in the manipulation of such an expression?
No problem.Thanks in advance