Question

How do you determine the limit of `cos(x)` as n approaches `oo`?

Answer 1

There is no limit (the limit does not exist).

Explanation:

As `x` increases without bound it takes on values of the form `pik` infinitely many times for even integer `k` and infinitely many times for odd integer `k`.

Therefore, as `x` increases without bound, `cosx` takes on values of `1` and of `-1`, infinitely many times.

It is not possible that there is an `L` that `cosx` is within `epsilon = 1/2` of for all `x` greater than some specified `N`.
Less formally, there is no `L` that all of the values of `cosx` are eventually close to.
Even less formally, the values of `cosx` cannot zero in on a particular `L` while also going all the way from `-1` to `1` and back again.

Finally, here is the graph:

graph{cosx [-9.75, 82.75, -22.43, 23.84]}