1: When n is large, the expression
How can this be proved? To do the above reasoning formally, one should factor out the dominant terms
Then it only remains to prove that the fractions in the last step indeed go to zero, which can be done by the l'Hospital rule (the fractions are of the indeterminate type).
Alternative solution would use the fact that the original ratio (after getting rid of the absolute value) is also of the type infinity over infinity, so the l'Hospital rule can be applied directly to it. Of course, this rule cannot be applied to powers, only fractions, but fortunately exponential is a nice function and as such it can be pulled out of the limit. Then one should remember to put the result (limit of the ratio) back to the exponential.