Evaluate:
There are eight million ways to do these, Ill show a couple. First let us handle
Way One: if we let this limit transforms to , now by the connection of the Root and Ratio tests, this is equivalent to
Way Two: After the sub we may rewrite it as and by the continuity of the exponentiat function across its entire domain we may rewrite this as . Now by any number of methods (I leave this to you) so we have our limit is equal to
Now my fingers are getting tired so lets only consider one method for this one. Once again let us rewrite our limit as
Now and
So we have that and that makes
So the asnwer to your actual limit is