Show that a function oscillates between two points?

Question

For the function $\text{cos} (x),$ how do I show a function oscillates between two points? This question was posed to me as part of a kinematics question in differential calculus. I know how to show the period of $\text{cos} (x)$ is $2\pi$ but I know showing the period is not enough. Should I find the maximum, $y = 1,$ and minimum, $y= -1,$ and show they are periodic also?

Answer 1

I don't know if this will answer your question, but if you're asking how to show that the function $\cos(x)$ has no limit and takes the values $1, -1$ infinite times you can start by noticing that, since the function is periodic (and non-constant) it has no limit; than you observe that $\cos(x)=1\implies x=2k\pi$ and $\cos(x)=-1\implies x=\pi+2k\pi$ where $k$ can be replaced by any natural number: so $\cos(x)$ takes these values (that are maximum and minimum of $\cos(x)$) infinite times, and you can state that your function oscillates between two points.