How do we use the result to find the limit?

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I know that for the first part of b:

lim_(n->∞)2^(1/n) = 2^(ln_(n->∞)(1/n)= 2^0 =1

However, I tried using a similar technique for the next part for a_n but it doesn't seem to work.

Please help

1 Answer
May 6, 2018

lim_(n->oo) (4^n+5^n)^(1/n) = 5

Explanation:

Note that:

5 = (5^n)^(1/n) < (4^n + 5^n)^(1/n) <= (2 * 5^n)^(1/n) = 5 * 2^(1/n)

So:

5 = lim_(n->oo) (5^n)^(1/n) <= lim_(n->oo) (4^n+5^n)^(1/n) <= lim_(n->oo) 5 * 2^(1/n) = 5