real analysis - Why are there limits which do not exist?

The limit of a real function can be of one of three types.

A real limit would be a limit where the value is a real number:
$$\lim_{x\to0}x^2=0$$
An infinite limit would be a limit where the value is equal to $\pm\infty$:
$$\lim_{x\to-\infty}{1\over e^x}=\infty$$

And an undefined/non-existing limit would be one where it is not possible to assign a value of any kind to the limit:
$$\lim_{x\to\infty}\sin x$$

My question is, what other undefined limits are out there? I understand that $\lim_{x\to\infty}\sin x$ is undefined because $\sin x$ is a periodic function whose value is in $[-1,1]$, and we can't find which, if any, single value in that interval it will achieve as $x$ tends to infinity. I also know I could get a similar undefined limit using any other trigonometric function.

Are there any other limits composed of elementary functions which are undefined? Are such undefined limits something exclusive to periodic functions or can they be constructed using other kinds of functions?

Answer

The limits that do not exist can be seen as the set of accumulation points.

In this sense, $\displaystyle\lim_{x\to\infty}\sin x = [-1,1]$ because for each $L\in [-1,1]$ there is a sequence $x_n \to \infty$ such that $\sin(x_n) \to L$.

So, more precisely, the set of accumulation points is the set of all possible limits that do exist.