Find lim(n->infinity) (3 + sin(n))^(1/(2n))?

Question

Thanks!

Answer 1

You can almost use the other answer... but you need one step first:

2 ≤ 3 + sin(n) ≤ 4

so

2^(1/(2n)) ≤ (3+sin(n))^(1/(2n)) ≤ 4^(1/(2n))

Now applying the limit and noting that lim(n->inf) 1/(2n) = 0 so the 2^(1/(2n)) becomes 2^0 = 1 (likewise with the 4^(1/(2n)) ), we find that

1 ≤ lim(n->inf) (3+sin(n))^(1/(2n)) ≤ 1

which shows that the limit is 1.

Answer 2

1/infinity=0 Anything to the 0 power =1. There you go.