If the limit can be computed by L'Hopital then it exists.
It's easy to find the limit by L'Hopital's rule, but that's the second part of the question after proving the limit exists.Verify the following limit exists:
Therefore, is there a way to prove that the limit exists without using L'hopitals rule (or is it acceptable to just find the limit by L'Hopitals rule and say that it must exist since we found it!)?
Hence your limit is equivalent to :
now simplify by x :
now you can see that the numerator simplifies to
and then you have the limit.
as to prove why it exists, I'm not sure, maybe finding the limit is enough??
Maybe you can just say that for small x. Thus It's crafty but maybe it works ?
there surely are some mistakes in the wording... It's not my favourite stuff, so you'll do it yourself
Let me do the same computation as you did, but with the o(): we have (or , the conclusion is the same), so that
if we divide by we get . This is the sum of a constant and a function which is negligible compared to . Since , we can't conclude anything. In fact, when I wrote , it was equivalent to write since the first term is negligible compared to .
The conclusion is that the expansion at the beginning is not precise enough (the neglected terms become important in the final expression). It would have been necessary to write . Then the numerator becomes:
and dividing by we get : the limit is .
" " is a function which is negligible compared to , so that in the expansion each term is negligible compared to the previous ones. This shows that the constants are the "right" ones: , ,... (provided the aren't 0).