calculus 2?

Question

Can anyone help me with the Evaluate the limit question. Ive tried it but its always wrong

Answer 1

The nth root is the 1/n power, so:

    a_n ^ (1/n) = [8 (1 + 1/n)^(n^2)] ^ (1/n)

                      = 8^(1/n) * (1 + 1/n)^n

And the limit of that as n-->oo is the product of the two limits 8^(1/n) --> 1 and (1 + 1/n)^n --> e.  So a_n^(1/n) --> 1*e = e as n-->oo.

Answer 2

(8 * (1 + 1/n)^(n^2))^(1/n) =>

8^(1/n) * (1 + 1/n)^(n^2 / n) =>

8^(1/n) * (1 + 1/n)^n

As n goes to infinity:

8^(1/n) goes to 1

(1 + 1/n)^n goes to e

1 * e =>

e