I want to find out $\displaystyle\lim_{x \to +\infty} x \sin x$. Now, this doesn't exist, but I'm not sure how to transform the definition of limit to something that lets me prove that the limit doesn't exist. This is the definition I use, for the record:
We say that $\displaystyle\lim_{x \to +\infty} f(x) = l$ if $\forall \epsilon > 0, \exists M > 0$ such that $x > M \implies |f(x)-l| < \epsilon$.
This isn't exactly $\epsilon-\delta$, it's more like $\epsilon - M$, but it's the same idea. My problem is: how to use this to prove that the limit doesn't exist? I know that I would have to begin like this:
We say that $\displaystyle\lim_{x \to +\infty} f(x)$ doesn't exist if $\exists \epsilon>0$ such that $\forall M > 0$ . . .
And I don't know how to continue.
Edit: I want to clarify something: while I am indeed trying to prove the nonexistence of $\displaystyle\lim_{x \to +\infty} x \sin x$, the point of this question was to be able to use the definition to prove the nonexistence of any limit, not just this one.