Question

What are some examples in which the limit does not exist?

Answer 1

One example is when the right and left limits are different. So in that particular point the limit doesn't exist.
For example: if you compress a gas the volume abruptly changes (in changing phase):

You can have a limit for `p` approaching `100` torr from the left (`=0.8 l`) or right (`0.3 l`) but not in `p=100` torr.
So: `lim_(p->100)V=` doesn't exist.

Here, to recapitulate, some interesting cases: