In the following problems, find the limit of the given sequence as n -> ∞?

Question

In the following problems, find the limit of the given sequence as n -> ∞.

1. 2^n / n^2

2. n sin (1/n)

Please explain your work. Thank you! :)

Answer 1

1. Let f(n) = 2^n/n^2. Then f(n+1) = 2^(n+1) / (n^2 + 2n + 1).

f(n+1)/f(n) = 2 n^2/(n^2+ 2n + 1) -> 2 as n -> ∞.

Ie. f(n) gets roughly twice as big every time n increases by 1, so f -> ∞.

2. As x -> 0, (sin x)/x -> 1

Plug in x = 1/n -> n sin(1/n) = (sin (1/n))/(1/n) -> 1