How do you know a limit does not exist?
In short, the limit does not exist if there is a lack of continuity in the neighbourhood about the value of interest.
Recall that there doesn't need to be continuity at the value of interest, just the neighbourhood is required.
Most limits DNE when
A common misunderstanding is that limits DNE when there is a point discontinuity in rational functions. On the contrary, the limit exists perfectly at the point of discontinuity!
So, an example of a function that doesn't have any limits anywhere is