Question

`lim_(n->oo)(Sin n+cos n)^n` Find the value? Thanks

Answer 1

See below.

Explanation:

`sinn+cosn = sqrt2 sin(n+pi/4)`

`(sinn+cosn)^n = 2^(n/2) sin^n(n+pi/4)`

but `(n+ pi/4)/(2pi)` is irrational so `a_n = sin(n+pi/4)` is not periodic and can take any value in `[-1,1]` and then concluding

`lim_(n->oo)(sinn+cosn)^n` does not exists.

Answer 2

The limit doesn't exist.

Explanation:

Since `sin` and `cos` are periodic functions, the limits as `n` approaches infinity of `sin n` and `cos n` `color(red)("never")` converge to a value.