How do we use the result to find the limit?

Question

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

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Answer 1 · 2018-05-06T07:31:34

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

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#