Question

use an appropriate method to evaluate the limit, if it exists, of each of the following sequences {ai}, If it is divergent explain why

Answer 1

\[[\cos\frac{ 1 }{ i }] ^{i^2}\]

Answer 2

let \[y=\lim_{n\rightarrow \infty}\left[\cos \frac{1}{n}\right]^{\large{n^2}}\]

consider \[\ln y = \lim_{n\rightarrow \infty} n^2\ln\left[\cos \frac{1}{n}\right]= \lim_{n\rightarrow \infty} \frac{\ln\left[\cos \frac{1}{n}\right]}{1/n^2}\]

Answer 3

L'Hopital's rule comes in handy in evaluating the limit.