calculus 2?

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

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  • 2 months ago

    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.

  • (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

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