This might be very simplistic, but could you use l'Hospital's Rule? It would certainly work for the real variable case, and would give you, as you intimated, 1/2.
Evaluate lim sqrt(n)*[sqrt(n+1)-sqrt(n)]
I know that limit sqrt(n) = Infinity and that limit (sqrt(n+1)-sqrt(n)) = 0. And I know that sqrt(n)*(sqrt(n+1)-sqrt(n)) = sqrt(n)/(sqrt(n+1)+sqrt(n)).
I believe the limit of the product of these sequences is 1/2, but I am not sure how to get there. Thanks for your help.
In that case, go back to Plato's formula: and try to find N such that for a given and if , then,
i.e. (note that we remove the absolute value signs - this quantity is positive!).
Now, this means: for large (check this!)
Therefore:
Hence:
You found your N!
Note that you don't have to find the best value of N. Any value that works if fine to prove the limit.