Let $f$ be a function $\mathbb R \to \mathbb R$. According to Wikipedia an discontinuity of $f$ is essential if and only if either the left or the right limit is infinite or does not exist.
Is it possible to construct an undefined non-infinite functional limit? (If so, could you show me such an example please)
I'm not sure it's correct but until now I thought of non-existing limits as ones that are $\pm \infty$. But maybe this was wrong and a limit of the kind $(-1)^n$ could be constructed for a function also.
A simple example does probably not work: I believe that if we define $f$ to be zero everywhere except $f({1 \over n}) = (-1)^n$ then the limit of $f$ at $0$ would probably be zero because $f$ is mostly zero around zero. Right?
The standard example is $$ f(x)=\sin\frac1x,\quad x\ne0. $$ As $x\to0$, $f(x)$ oscillates between $1$ and $-1$.