thus by the squeeze theorem:
Squeeze theorem? I've not heard of that, but I was under the impression that I was suppose to solve the question using l'Hopital's rule, except I kept getting the limit of the derivatives to be 0 or infinity and the whole limit is still indeterminate. Does that mean I can't use that rule here?
Stop thinking mechanically, first look at the problem at hand and think.
As you can see this is a limit of a sequence - a discrete set of values. The derivative is a continues variable operator, whereas in this case we're given a sequence - which is a discrete set of values, so the derivative has absolutely no meaning when applied to the series. This is exactly what happens when people try to apply theorems, operators, transformations... without understanding their meaning and the necessary conditions for their application. I can't blame you because I was too one of those people, but I've learned the hard way that I had to change.
We could apply the discrete version of L'Hospital rule to the sequence (The derivative is replaced with a difference), but unfortunately the necessary conditions do not hold.
Squeeze theorem - Wikipedia, the free encyclopedia
Sorry everyone! Once again my famous carelessness has caused a problem. Let me be more explicit.
This is no proof but it gets the idea off. Consider
It can be seen that this is an infinity over infinity case and we can apply L'hospitals. In fact it will be infinity over infinity until the a+1th derivative. In other words
Therefore
And if your grievance was that I had a>1 and not |a|>1 just make and use the Squeeze theorem to deduce the limit is still zero.