# Disproving a limit of composite functions using Epsilon proof

I've seen lots of questions demonstrating how to prove a limit doesn't exist, however there aren't any solutions using composite functions and limits at infinity. The question I'm trying to solve is as follows:

Let f: [-1, 1] --> R be a function. Suppose that f is not a constant function. Define g: R --> R by g(x) = f(sin(x)). Prove that limx→∞ g(x) does not exist.

As $$f$$ is non-constant, it will take at least two different values. Then $$g$$ alternates periodically between them and cannot converge.
• @S.Miller: take $\epsilon$ smaller than half the difference between the values mentioned in my answer.