compute the limn->∞sup(((3^n+(-2)^n)^(1/n))/n^(1/n))

We can start by using the following inequality for any positive real numbers a and b:

(a + b)^(1/n) ≤ a^(1/n) + b^(1/n)

We can apply this inequality to the expression inside the limit as follows:

(3^n + (-2)^n)^(1/n) = [(3^n)^(1/n) + ((-2)^n)^(1/n)]/2 ≤ [3 + (-2)]/2 = 1

Dividing both sides by n^(1/n), we get:

[(3^n + (-2)^n)^(1/n)]/n^(1/n) ≤ 1/n^(1/n)

Taking the limit as n approaches infinity, we see that the right-hand side goes to zero:

lim n->∞ (1/n^(1/n)) = 0

Therefore, by the squeeze theorem, the limit of the original expression is also zero:

lim n->∞ sup[((3^n + (-2)^n)^(1/n))/n^(1/n)] ≤ lim n->∞ (1/n^(1/n)) = 0

Hence, the desired limit is 0.