What is the value of $\sin(x)$ if $x$ tends to infinity? 
What is the value of $\sin(x)$ if $x$ tends to infinity?

As in wikipedia entry for "Sine", the domain of $\sin$ can be from $-\infty$ to $+\infty$. What is the value of $\sin(\infty)$?

Mathematics has many different ways of talking about infinity. In particular, there is more than one way of adjoining infinite quantities to the real number line. One, which is similar in spirit to what  the others have been talking about, is the extended real number line. Another is the hyperreals. In the hyperreals, we have many different sizes of infinity, functions such as the sine can be extended to the whole hyperreal line in a natural and uniquely defined way, and it makes sense to say that $\sin x$ has some value, where $x$ is a certain infinite number. There is also a notion of an infinite integer, and we can, for example, say that $\sin(\pi n)=0$ and $\sin(\pi (n+1/2))=\pm 1$ if $n$ is an infinite integer.
What this still does not allow is any conclusion about the value of $\sin x$ as $x$ tends to infinity: $\sin x$ takes on all values between $-1$ and $+1$ for various infinite values of $x$.
