Why is $\cos(\infty)$ undefined?

Question

Why is $\cos(\infty)$ undefined?

I really don't understand this. Is it because we can't pinpoint an exact value for cos at infinity?

Answer 1

You almost understand it.

We don't need to pinpoint an exact value, but we do need to pinpoint a value, say $L$, that, as $x$ gets larger, then the value of $\cos x$ gets closer to $L$. Since $\cos x$ is always wiggling between $-1$ and $1$, we can't do that.

Answer 2

because you cannot attribute it a value. The value of $cos(\infty)$ would be defined as: $$ cos(\infty)=\lim_{x\rightarrow\infty}cos(x) $$ If this limit was well defined then for any sequence $a_n\rightarrow \infty$ one would have: $$ cos(\infty)=\lim_{n\rightarrow\infty}cos(a_n) $$ giving the same value.

However if you consider $a_n=2n\pi$ and $b_n=(2n+1)\pi$ you get: \begin{align*} cos(\infty)&=\lim_{n\rightarrow\infty}cos(a_n)=+1 \\ &=\lim_{n\rightarrow\infty}cos(b_n)=-1 \end{align*} As these values are differents, the limit does not exist, thus $cos(\infty)$ is not defined.