What does it mean to say that f(3) exists but that there is no limit to f(x) as x approaches 3?
Does that count?
If I remember correctly, in real analysis we proved that $\displaystyle \lim_{x \to 0} \sin(1/x) $ does not exist by finding a sequence $\displaystyle (s_n)$ that converges to some value c ($\displaystyle s_{n} \ne c$) but where $\displaystyle f(s_n)$ doesn't converge (or something like that).