Evaluate the limit.
First, we want to get expansions for and as . For we write and use the expansion (page 287 of Apostol) of . This gives us
Next, for , again we write and then use the expansion for we have
Now, we need use the expansion for (again, page 287 of Apostol)
and substitute this into our expansion of ,
(Again, this is the really nice part of little -notation. We had lots of terms in powers of greater than 3, but they all get absorbed into , so we don’t actually have to multiply out and figure out what they all were. We only need to figure out the terms for the powers of up to 3. Of course, the 3 could be any number depending on the situation; we chose 3 in this case because we know that’s what we will want in the limit we are trying to evaluate.)
So, now we have expansions for and (in which most of the terms cancel when we subtract) and we can evaluate the limit.