Obviously, if $f(x)$, $x \to \infty$, has a finite limit, the sequence $a_n = f(n)$ converges. The converse does not hold ($\sin \pi x)$. A sufficient condition for the converse to hold is the monotonicity of $f$ (I think).
What's a condition which is both necessary and sufficient?
In other words, how do we characterize those functions for which
$(1) $ The limit of the sequence and function both exist.
$(2) \lim\ f(x) = \lim \ f(n)$